This document provides an overview of the origin and history of continuous and discrete convolution operations. It discusses how the real convolution integral first occurred in works related to series expansion and Taylor series by mathematicians like D'Alembert and Laplace in the 18th century. It also describes how the real convolution integral was used by Fourier in the early 19th century to represent functions by Fourier series. Later, mathematicians like Dirichlet, Riemann, and Weierstrass also employed real convolution integrals in their studies of Fourier series. The document is divided into multiple chapters that each trace the use of the real convolution integral in different areas of mathematics throughout history.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
Solved numerical problems of fourier seriesMohammad Imran
This document is a report by Mohammad Imran on solved numerical problems of Fourier series. It discusses Fourier series and provides solutions to questions involving Fourier series. The report is presented to the Jahangirabad Institute of Technology as part of a semester 2 course on the topic of Fourier series.
This document summarizes the Runge-Kutta methods for solving differential equations numerically. It introduces the first, second, third, and fourth order Runge-Kutta methods and provides the equations for calculating each. An example of using the fourth order Runge-Kutta method to solve the differential equation dy/dx=x+y is shown step-by-step. The example calculates the solution to y(0.2) given y(0)=1 using increments of h=0.1.
This document discusses Fourier series and related concepts. It provides definitions and formulas for general Fourier series, Fourier series for discontinuous functions, the change of interval method, Fourier series for even and odd functions, and half range Fourier cosine and sine series. Examples of applications of these Fourier series concepts and techniques are also presented.
This document summarizes and compares several numerical methods for solving ordinary differential equations (ODEs):
- Euler's method approximates the tangent line at each step to find successive y-values. While simple, it has local truncation errors that accumulate.
- Improved Euler's method takes the average slope between the current and next steps to give a more accurate approximation.
- Runge-Kutta methods such as the fourth-order method provide much greater accuracy than Euler or improved Euler by using multiple slope estimates within each step.
An example applies each method to the ODE dy/dx = x + y to compare their results in solving for successive y-values out to x = 0.3.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
Solved numerical problems of fourier seriesMohammad Imran
This document is a report by Mohammad Imran on solved numerical problems of Fourier series. It discusses Fourier series and provides solutions to questions involving Fourier series. The report is presented to the Jahangirabad Institute of Technology as part of a semester 2 course on the topic of Fourier series.
This document summarizes the Runge-Kutta methods for solving differential equations numerically. It introduces the first, second, third, and fourth order Runge-Kutta methods and provides the equations for calculating each. An example of using the fourth order Runge-Kutta method to solve the differential equation dy/dx=x+y is shown step-by-step. The example calculates the solution to y(0.2) given y(0)=1 using increments of h=0.1.
This document discusses Fourier series and related concepts. It provides definitions and formulas for general Fourier series, Fourier series for discontinuous functions, the change of interval method, Fourier series for even and odd functions, and half range Fourier cosine and sine series. Examples of applications of these Fourier series concepts and techniques are also presented.
This document summarizes and compares several numerical methods for solving ordinary differential equations (ODEs):
- Euler's method approximates the tangent line at each step to find successive y-values. While simple, it has local truncation errors that accumulate.
- Improved Euler's method takes the average slope between the current and next steps to give a more accurate approximation.
- Runge-Kutta methods such as the fourth-order method provide much greater accuracy than Euler or improved Euler by using multiple slope estimates within each step.
An example applies each method to the ODE dy/dx = x + y to compare their results in solving for successive y-values out to x = 0.3.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
This document provides historical context on key concepts in Schwartz space and test functions. It discusses how Laurent Schwartz defined the Schwartz space in 1947-1948 to consist of infinitely differentiable functions that, along with their derivatives, decrease faster than any polynomial. Test functions, a subset of Schwartz space, have compact support. Joseph-Louis Lagrange and Norbert Wiener helped develop the method of multiplying a function by a test function and integrating, which is fundamental to distribution theory. The term "mollifier" for test functions was coined by Kurt Friedrichs in 1944, although Sergei Sobolev had previously used them. Many mathematicians, including Leray, Sobolev, Courant, Hilbert, and Weyl,
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
This document discusses Taylor series expansions. It defines Taylor series as the expansion of a complex function f(z) that is analytic inside and on a simple closed curve C in the z-plane. The Taylor series expresses f(z) as a power series centered at a point z0 within C. It provides examples of standard Taylor series expansions and worked illustrations of expanding various functions as Taylor series. The document also notes that the radius of convergence of a Taylor series is defined by the distance to the nearest singularity from the center point z0.
This document discusses Newton's forward and backward difference interpolation formulas for equally spaced data points. It provides the formulations for calculating the forward and backward differences up to the kth order. For equally spaced points, the forward difference formula approximates a function f(x) using its kth forward difference at the initial point x0. Similarly, the backward difference formula approximates f(x) using its kth backward difference at x0. The document includes an example problem of using these formulas to estimate the Bessel function and exercises involving interpolation of the gamma function and exponential function.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
Topic: Fourier Series ( Periodic Function to change of interval)Abhishek Choksi
The document discusses Fourier series and their properties. Fourier series can be used to represent periodic functions as an infinite sum of sines and cosines. The key points are:
- Fourier series can represent functions over any interval length by transforming the variable.
- Examples show how to calculate the Fourier coefficients for specific functions over given intervals.
- The Fourier series representation allows periodic functions to be broken down into their constituent trigonometric components.
This document provides an overview of Fourier series and Fourier transforms. It discusses the history of Fourier analysis and how Fourier introduced Fourier series to solve heat equations. It defines Fourier series and covers topics like odd and even functions, half-range Fourier series, and the complex form of Fourier series. The document also discusses the relationship between Fourier transforms and Laplace transforms. It concludes by listing some applications of Fourier analysis in fields like electrical engineering, acoustics, optics, and more.
Newton divided difference interpolationVISHAL DONGA
This document presents Newton's divided difference polynomial method of interpolation. It defines interpolation as finding the value of 'y' at an unspecified value of 'x' given a set of (x,y) data points. Newton's method uses divided differences to determine the coefficients of a polynomial that can be used to interpolate and estimate y-values between the given data points. The document includes an example of applying Newton's method to find the interpolating polynomial and estimate an unknown y-value for a given set of 5 (x,y) data points.
This document discusses the discrete-time Fourier transform (DTFT). It begins by introducing the DTFT and how it can be used to represent aperiodic signals as the sum of complex exponentials. Several properties of the DTFT are then discussed, including linearity, time/frequency shifting, periodicity, and conjugate symmetry. Examples are provided to illustrate how to compute the DTFT of simple signals. The document also discusses how the DTFT can be used to represent periodic signals and impulse trains.
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
The document discusses the inverse Laplace transform and related topics. It provides three main cases for performing partial fraction expansions when taking the inverse Laplace transform: 1) non-repeated simple roots, 2) complex poles, and 3) repeated poles. It also discusses the convolution integral and how it relates the time domain convolution of two functions to the multiplication of their Laplace transforms. An example uses the convolution integral to find the output of a system given its impulse response and input.
Dcs lec03 - z-analysis of discrete time control systemsAmr E. Mohamed
The document discusses discrete time control systems and their mathematical representation using z-transforms. It covers topics such as impulse sampling, the convolution integral method for obtaining the z-transform, properties of the z-transform, inverse z-transforms using long division and partial fractions, and mapping between the s-plane and z-plane. Examples are provided to illustrate various concepts around discrete time systems and their analysis using z-transforms.
The document discusses periodic functions and their properties. The key points are:
- A periodic function f(x) satisfies f(x) = f(x + T) for some fixed period T and all real x.
- Periodic functions repeat their values at intervals of their period, including integer multiples of the period.
- Functions are defined as even if f(-x) = f(x) and odd if f(-x) = -f(x).
- Several important formulas are provided for integrating exponential and trigonometric functions.
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...saahil kshatriya
The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including:
1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a.
2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t).
3) An example calculates the Laplace transform of t2u(t-2) using the
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
Este documento describe la ingeniería social y cómo los hackers la usan para obtener información de seguridad de TI engañando al personal de una organización. Explica que la ingeniería social implica hacerse pasar por alguien de confianza para persuadir u obtener información del personal a través de preguntas. También recomienda que las organizaciones tomen medidas como establecer políticas de seguridad, capacitar a los empleados para reconocer tales tácticas, y contratar hackers éticos que ayuden a identificar y mitigar riesgos de ingen
1. The document discusses communication noise in project management environments. Noise arises from stakeholders expressing feelings about the project through opinions and rumors, which interferes with message transmission.
2. A basic communication model in project management includes senders, receivers, messages, and noise. Stakeholders are the primary senders. Noise occurs when stakeholders react negatively or positively to a project breaking established work rules.
3. Noise can have effects on project progress such as temporary disruptions, changes in trend direction or rate, or even trend disruption. The project manager must take actions to deal with noise and its potential bandwagon effects.
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
This document provides historical context on key concepts in Schwartz space and test functions. It discusses how Laurent Schwartz defined the Schwartz space in 1947-1948 to consist of infinitely differentiable functions that, along with their derivatives, decrease faster than any polynomial. Test functions, a subset of Schwartz space, have compact support. Joseph-Louis Lagrange and Norbert Wiener helped develop the method of multiplying a function by a test function and integrating, which is fundamental to distribution theory. The term "mollifier" for test functions was coined by Kurt Friedrichs in 1944, although Sergei Sobolev had previously used them. Many mathematicians, including Leray, Sobolev, Courant, Hilbert, and Weyl,
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
This document discusses Taylor series expansions. It defines Taylor series as the expansion of a complex function f(z) that is analytic inside and on a simple closed curve C in the z-plane. The Taylor series expresses f(z) as a power series centered at a point z0 within C. It provides examples of standard Taylor series expansions and worked illustrations of expanding various functions as Taylor series. The document also notes that the radius of convergence of a Taylor series is defined by the distance to the nearest singularity from the center point z0.
This document discusses Newton's forward and backward difference interpolation formulas for equally spaced data points. It provides the formulations for calculating the forward and backward differences up to the kth order. For equally spaced points, the forward difference formula approximates a function f(x) using its kth forward difference at the initial point x0. Similarly, the backward difference formula approximates f(x) using its kth backward difference at x0. The document includes an example problem of using these formulas to estimate the Bessel function and exercises involving interpolation of the gamma function and exponential function.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
Topic: Fourier Series ( Periodic Function to change of interval)Abhishek Choksi
The document discusses Fourier series and their properties. Fourier series can be used to represent periodic functions as an infinite sum of sines and cosines. The key points are:
- Fourier series can represent functions over any interval length by transforming the variable.
- Examples show how to calculate the Fourier coefficients for specific functions over given intervals.
- The Fourier series representation allows periodic functions to be broken down into their constituent trigonometric components.
This document provides an overview of Fourier series and Fourier transforms. It discusses the history of Fourier analysis and how Fourier introduced Fourier series to solve heat equations. It defines Fourier series and covers topics like odd and even functions, half-range Fourier series, and the complex form of Fourier series. The document also discusses the relationship between Fourier transforms and Laplace transforms. It concludes by listing some applications of Fourier analysis in fields like electrical engineering, acoustics, optics, and more.
Newton divided difference interpolationVISHAL DONGA
This document presents Newton's divided difference polynomial method of interpolation. It defines interpolation as finding the value of 'y' at an unspecified value of 'x' given a set of (x,y) data points. Newton's method uses divided differences to determine the coefficients of a polynomial that can be used to interpolate and estimate y-values between the given data points. The document includes an example of applying Newton's method to find the interpolating polynomial and estimate an unknown y-value for a given set of 5 (x,y) data points.
This document discusses the discrete-time Fourier transform (DTFT). It begins by introducing the DTFT and how it can be used to represent aperiodic signals as the sum of complex exponentials. Several properties of the DTFT are then discussed, including linearity, time/frequency shifting, periodicity, and conjugate symmetry. Examples are provided to illustrate how to compute the DTFT of simple signals. The document also discusses how the DTFT can be used to represent periodic signals and impulse trains.
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
The document discusses the inverse Laplace transform and related topics. It provides three main cases for performing partial fraction expansions when taking the inverse Laplace transform: 1) non-repeated simple roots, 2) complex poles, and 3) repeated poles. It also discusses the convolution integral and how it relates the time domain convolution of two functions to the multiplication of their Laplace transforms. An example uses the convolution integral to find the output of a system given its impulse response and input.
Dcs lec03 - z-analysis of discrete time control systemsAmr E. Mohamed
The document discusses discrete time control systems and their mathematical representation using z-transforms. It covers topics such as impulse sampling, the convolution integral method for obtaining the z-transform, properties of the z-transform, inverse z-transforms using long division and partial fractions, and mapping between the s-plane and z-plane. Examples are provided to illustrate various concepts around discrete time systems and their analysis using z-transforms.
The document discusses periodic functions and their properties. The key points are:
- A periodic function f(x) satisfies f(x) = f(x + T) for some fixed period T and all real x.
- Periodic functions repeat their values at intervals of their period, including integer multiples of the period.
- Functions are defined as even if f(-x) = f(x) and odd if f(-x) = -f(x).
- Several important formulas are provided for integrating exponential and trigonometric functions.
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...saahil kshatriya
The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including:
1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a.
2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t).
3) An example calculates the Laplace transform of t2u(t-2) using the
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
Este documento describe la ingeniería social y cómo los hackers la usan para obtener información de seguridad de TI engañando al personal de una organización. Explica que la ingeniería social implica hacerse pasar por alguien de confianza para persuadir u obtener información del personal a través de preguntas. También recomienda que las organizaciones tomen medidas como establecer políticas de seguridad, capacitar a los empleados para reconocer tales tácticas, y contratar hackers éticos que ayuden a identificar y mitigar riesgos de ingen
1. The document discusses communication noise in project management environments. Noise arises from stakeholders expressing feelings about the project through opinions and rumors, which interferes with message transmission.
2. A basic communication model in project management includes senders, receivers, messages, and noise. Stakeholders are the primary senders. Noise occurs when stakeholders react negatively or positively to a project breaking established work rules.
3. Noise can have effects on project progress such as temporary disruptions, changes in trend direction or rate, or even trend disruption. The project manager must take actions to deal with noise and its potential bandwagon effects.
Este documento discute la importancia de seguir métodos sistemáticos para el desarrollo de tecnología de información (TI), como identificar claramente los requisitos, analizar el problema, diseñar una solución, implementarla y darle seguimiento. Si no se siguen estas fases, es probable que el proyecto falle o tenga consecuencias negativas. También analiza el método común de "codificar y corregir", el cual omite estas fases sistemáticas y suele resultar en proyectos defectuosos. Finalmente, concluye
Este documento describe los diferentes modelos curriculares de posgrado en informática y computación. Explica brevemente el concepto de posgrado, sus objetivos y tipos. Luego, presenta la arquitectura de la informática y sus entidades. Finalmente, propone dos modelos curriculares de posgrado basados en estas entidades: uno organiza el plan de estudios en torno a las entidades y otro en torno a los nombres de los programas.
El documento proporciona recomendaciones para que los administradores de proyectos de TI eviten el colapso de los proyectos. Sugiere contar con un plan claro y definir requisitos y entregables para minimizar el impacto de los cambios, los cuales son inevitables. También recomienda documentar todos los cambios en el alcance para justificar las acciones tomadas y aprender de experiencias pasadas en caso de que el colapso sea inevitable.
Este módulo presenta los fundamentos para el desarrollo de proyectos informáticos. La sección 1 discute los cinco elementos clave para el desarrollo de proyectos: personas, información, procesos, herramientas y productos/servicios. La sección 2 analiza los modelos de madurez relacionados con las personas y los procesos. La sección 3 cubre los elementos de un plan de comunicación y estrategias de comunicación. El objetivo general es analizar estos componentes y su relación, con un enfoque en las personas y los proces
Las carreras en tecnologías de la información y telecomunicaciones tendrán un fuerte crecimiento, con un aumento estimado del 24% en puestos de trabajo en TI para el 2016. También habrá alta demanda para profesionales de la salud debido al envejecimiento de la población. Para tener éxito en el futuro, los estudiantes necesitarán desarrollar habilidades como la expresión oral y escrita, dominio de idiomas e innovación. Las carreras que combinan diferentes áreas de conocimiento, así como aquellas enfocadas
The document summarizes the spectral theory and principle of limiting absorption for the elasticity operator in infinite, homogeneous, anisotropic media. It defines the elasticity operator and establishes its properties, including that it is self-adjoint. It derives the wave equation for elastic media from Hooke's law and the equations of motion. It shows plane wave solutions to the wave equation and establishes Christoffel's equation.
Este documento discute la existencia de los "hackers éticos", personas que tienen las mismas habilidades técnicas que los hackers pero las usan para propósitos éticos como encontrar vulnerabilidades de seguridad en sistemas y redes para que las organizaciones puedan corregirlas. Los hackers éticos son valiosos para las organizaciones porque pueden simular ataques cibernéticos y elevar el nivel de seguridad. Algunas instituciones ofrecen certificaciones en hacking ético para expandir la forma de pensar de los ingenieros de sist
El documento describe la importancia de la gestión de riesgos en los negocios. Explica que la gestión de riesgos mejora las oportunidades de éxito de una empresa al identificar y minimizar las amenazas. También describe los pasos clave del proceso de gestión de riesgos, incluyendo establecer el contexto, identificar riesgos, analizar riesgos, evaluar riesgos, tratar riesgos, monitorear riesgos y comunicar sobre riesgos. Concluye que si una empresa no lleva a cabo la gestión de
La mejora de procesos en las empresas es importante para optimizar los procesos de negocio y alcanzar mejores resultados con una mejor utilización de recursos. La mejora de procesos debe llevarse a cabo como un proyecto utilizando métodos como definir el problema, identificar causas, seleccionar soluciones, e implementar y dar seguimiento a los cambios. Algunas consideraciones clave incluyen involucrar a las personas afectadas, proveer valor agregado, y mostrar liderazgo durante el proceso de mejora.
The document presents a model and set of policies for delivering education services based on the Zachman Architecture framework. The model identifies key entities involved in education services and defines them across rows. Question indicators like what, how, where, who, when and why are defined across columns to provide different views of education services. Filling the cells of the matrix provides a comprehensive description of education service delivery. The policies are then derived from the information in each cell to standardize education service implementation and operation both within and across institutions.
This document discusses fundamentals of information technology project development. It identifies the five key elements of IT project development as people, process, product/service, information, and tools. It emphasizes that successful projects require balancing these elements. The document also discusses two models for assessing project maturity: the Software Capability Maturity Model and the People Capability Maturity Model. Finally, it covers communication within projects and outlines elements of an effective communication plan and strategies.
Este documento discute las opciones educativas para que los emprendedores se mantengan actualizados en áreas fuera de su formación original. Señala cuatro tendencias clave en las que los emprendedores deben capacitarse constantemente: logística, internet, nuevas tecnologías de producción y sustentabilidad. También ofrece consejos como definir claramente las necesidades de capacitación, elegir programas alineados a los requerimientos del negocio y considerar opciones como maestrías, cursos o diplomados.
Este documento discute lo que significa ser un líder de proyectos. Explica que un líder de proyectos debe guiar a un equipo de trabajo hacia el éxito de un proyecto temporal, utilizando conocimientos y habilidades en gestión de proyectos. Sin embargo, el autor ha encontrado que algunas personas que se identifican como líderes de proyectos en realidad no cumplen con estos requisitos, careciendo de un proyecto, equipo, metodología o habilidades de comunicación. Para ser considerado un verdadero líder de
The document describes a competency-based human resources architecture for logistics enterprises. It defines five core competency components - logistics director, analyst, designer, supervisor, and operator - based on a logistics lifecycle model. For each component, it identifies the necessary skills and knowledge. The skills are defined based on technical vs. human relations requirements. The knowledge is defined using a modified Zachman framework tailored for logistics. The architecture was validated through a survey of Spanish logistics enterprises, with results showing varying levels of completeness across the components.
El documento habla sobre los hackers éticos. Explica que aunque tradicionalmente los hackers han sido vistos como delincuentes, existen ahora los llamados "hackers éticos" que trabajan con el permiso de las organizaciones para encontrar vulnerabilidades de seguridad en sus sistemas y redes. Estos hackers éticos son considerados una parte importante de la seguridad de muchas organizaciones y países. El documento concluye que aunque algunos niegan la existencia de los hackers éticos, estos existen y son valiosos para las organizaciones al pensar como hackers pero
Este documento analiza el efecto de una alerta epidemiológica en las empresas y propone estrategias para la recuperación empresarial. La alerta por el virus A-H1N1 ha interrumpido muchas actividades y afectado a las personas, que son el principal activo de las empresas. Para una recuperación eficaz, las empresas deben monitorear información oficial, evaluar daños financieros, identificar procesos críticos y desarrollar nuevos productos/servicios mediante proyectos bien administrados. Siguiendo estas estrategias de
A colleague of yours has given you mathematical expressions for the f.pdfarjuntiwari586
The document analyzes and compares different mathematical expressions for calculating electromagnetic fields given time-dependent charge and current distributions. It summarizes a derivation showing that an expression given by Panofsky and Phillips (involving retarded potentials) can be transformed into a form that better highlights the transverse nature of radiation fields. It also clarifies why a term in the electric field expression vanishes in the static case.
The document provides an overview of Fourier series. Key points include:
- Fourier series decompose periodic functions into the sum of sines and cosines, allowing representation as a series.
- They are named after Jean-Baptiste Joseph Fourier who introduced them to solve heat equations.
- Fourier series have advantages like representing discontinuous functions and periodic phenomena with a linear system of harmonics.
- The coefficients of the Fourier series define the contribution of each harmonic term to the overall function representation.
This seminar will cover the basics of complex manifolds and geometry through readings and presentations on topics from reference book [2]. The seminar will be organized as a reading group, with participants reading assigned material each week and one person giving a talk on a particular topic. Some of the topics to be covered include holomorphic functions, complex manifolds, vector bundles, divisors, projective space, differential forms, Kähler manifolds, and applications to curves.
The document discusses relations and functions. It defines a relation as a set of ordered pairs, with the domain as the set of first coordinates and the range as the set of second coordinates. A function is a special type of relation where each element of the domain is paired with exactly one element of the range, or no two ordered pairs have the same first coordinate. Examples are provided to illustrate relations, identifying their domains and ranges, and to demonstrate the vertical line test for determining if a relation is a function.
The document discusses the history and development of Taylor series. Some key points:
1) Brook Taylor introduced the general method for constructing Taylor series in 1715, after which they are now named. Taylor series represent functions as infinite sums of terms calculated from derivatives at a single point.
2) Special cases of Taylor series, like the Maclaurin series centered at zero, were explored earlier by mathematicians like Madhava and James Gregory.
3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.
1) Fourier developed a theory of infinite series of sine and cosine functions to solve differential equations describing vibratory motion that had no solutions in terms of elementary functions. These series, now known as Fourier series, can represent any periodic function.
2) A Fourier series decomposes a periodic function into an infinite sum of sines and cosines of integer multiples of the fundamental frequency. The coefficients of the sines and cosines are determined from the function itself.
3) Under certain conditions on the periodic function and its derivatives, the Fourier series converges to the value of the function at points of continuity and averages the left and right limits at points of discontinuity.
This presentation introduces Taylor series. It begins with background on Brook Taylor, who formally introduced Taylor series in 1715, and discusses their history. The presentation defines Taylor series as representing a function as an infinite sum of terms calculated from the function's derivatives at a single point. Examples are provided of using Taylor series to approximate functions and solve otherwise difficult problems in restricted domains. Applications of Taylor series mentioned include finding sums of series, evaluating limits, and approximating polynomial functions. The presentation concludes by thanking the audience and asking for any questions.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
This document provides an introduction to Fourier series. It lists the names and student IDs of the presenters. It then discusses Joseph Fourier, the founder of Fourier series, and provides definitions and properties of Fourier series including that they can represent both continuous and discontinuous periodic functions using sine and cosine terms. Examples of functions that can and cannot be represented by Fourier series are given. The document also discusses applications of Fourier series to problems involving heat transfer and vibrating systems.
The document discusses coherence in light sources and its impact on interference patterns observed using a Michelson interferometer. It introduces temporal coherence and coherence time, which describe the stability of the light wave's phase over time. Sources with a narrow spectral width like lasers have high coherence, while broad-spectrum sources have low coherence and do not produce clear interference patterns when the path length difference exceeds the coherence length. The visibility of interference fringes is directly related to the first-order coherence function, which quantifies how the wave's phase correlates over time.
This document provides an overview and forward for a book on the analysis, synthesis, and optimization of kinematic chains. It discusses the motivation for writing the book, which was to systematically compile and establish the foundations of existing knowledge on linkages. The book aims to present linkage modeling, analysis, synthesis, and optimization in a unified way based on rigorous theoretical foundations. It also discusses departures taken from other works, such as representing transformations with separate rotation and translation components instead of combined transformation matrices. The forward acknowledges other relevant works published since the book's conception and explains why publication of this text is still valuable for providing a new perspective and unified treatment of the subject from fundamentals to applications.
This document provides an introduction to general relativity. It begins by summarizing the key aspects of special relativity, including that spacetime is four-dimensional and transformations between inertial frames form the Poincare group. It then discusses the equivalence principle and introduces curved coordinates to describe gravity. The document derives the affine connection and Riemann curvature tensor, and introduces the metric tensor. It provides the perturbative expansion leading to Einstein's field equations and discusses solutions like the Schwarzschild metric and gravitational radiation.
Master Thesis on Rotating Cryostats and FFT, DRAFT VERSIONKaarle Kulvik
This thesis studied the vibrational and rotational aspects of the Cryo I Helsinki cryostat using Fourier analysis methods. Extensive Fourier analysis was performed to model the vibrations mathematically. The goal was to lower noise levels to improve cryostat operations. The second part of the work tested the homogeneity of a superconducting magnet. Significant preparation was required to build testing equipment for evaluating the magnet.
Fourier transforms and series are used in many aspects of geodynamics. They are used to solve linear partial differential equations, design antennas, process and analyze geophysical data, and represent functions on a spherical earth. The discrete Fourier transform allows representation of functions over a finite area with a discrete set of wavenumbers and Fourier coefficients. Properties like the shift and differentiation properties allow simplifying differential equations into algebraic equations.
The document is Ivan Martino's 2014 doctoral thesis from Stockholm University titled "Ekedahl Invariants, Veronese Modules and Linear Recurrence Varieties". It contains four papers at the intersection of algebraic geometry, commutative algebra, and combinatorics. The first two papers concern new geometric invariants for finite groups called Ekedahl invariants. The third paper discusses a combinatorial approach to studying Veronese modules. The fourth paper shows a recent result relating linear recurrence varieties to numerical semigroups.
The document provides an overview and history of the wavelet transform. It can be summarized as follows:
1. The wavelet transform was developed to address limitations of the Fourier transform and short-time Fourier transform in analyzing signals both in time and frequency. It uses wavelets of limited duration that can be scaled and translated.
2. The history of the wavelet transform began in 1909 with Haar wavelets. The concept of wavelets was then proposed in 1981 and the term was coined in 1984. Important developments included the construction of additional orthogonal wavelets in 1985, the proposal of the multiresolution concept in 1988, and the fast wavelet transform algorithm in 1989, enabling numerous applications.
3.
This document discusses uncertainty principles and their application to the double slit experiment. It summarizes Heisenberg's uncertainty principle and its limitations in describing position and momentum spreads. It then applies various uncertainty inequalities to analyze Bohr's argument that an interference pattern requires not knowing which slit particles pass through. Local uncertainty principles assert that low momentum uncertainty implies not only large position uncertainty, but low probability of localization. The document analyzes applying these principles to justify Bohr's response to Einstein's proposed resolution of the double slit ambiguity.
FRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHIMILANJOSHIJI
This document provides an introduction to fractal geometry and its applications. It discusses how fractals were discovered by Mandelbrot as a way to describe irregular patterns in nature. Some key areas where fractals are applied that are mentioned include astronomy (galaxies and Saturn's rings), biology (bacteria cultures, plants), data compression, diffusion, economy, special effects in movies like Star Trek, weather patterns, antennas, and understanding global warming. The document also provides mathematical definitions of fractals and the Mandelbrot set, and describes how to graph the Mandelbrot set. Examples of fractals discussed in more depth include the Sierpinski triangle, Koch curve, broccoli, veins, and the Lorenz
This document provides an introduction to fractal geometry and its applications. It discusses how fractals were discovered by Mandelbrot as a way to describe irregular patterns in nature. Some key areas where fractals are applied that are mentioned include astronomy (galaxies and Saturn's rings), biology (bacteria cultures, plants), data compression, special effects in movies like Star Trek, weather patterns, heartbeats, antennas, and modeling global warming. The document also provides mathematical definitions of fractals and the Mandelbrot set, and describes how to generate fractal landscapes and the Mandelbrot set using iterated functions.
El documento proporciona orientación sobre cómo elegir una maestría. Sugiere considerar tu licenciatura, experiencia profesional, posición actual y deseada, así como las necesidades de las empresas. Recomienda elegir una maestría que complemente tus conocimientos y experiencia, permita interactuar con profesionales, y vincule lo aprendido con la práctica laboral.
El documento analiza la estrategia de Wile E. Coyote para atrapar al Correcaminos, la cual incluye definir claramente su objetivo, prepararse estudiando, planear cuidadosamente sus acciones, simular y mejorar sus planes antes de ejecutarlos repetidamente hasta alcanzar su objetivo con perseverancia. Aunque al final logre atraparlo, Wile E. Coyote sabe que también debe estar preparado para lidiar con el éxito y sus posibles consecuencias.
El documento discute los problemas actuales en la educación y el trabajo en México. Señala que hay una vinculación entre la educación y el trabajo, de modo que los problemas en uno tienen consecuencias en el otro. Aunque ha habido avances en la educación mexicana, la calidad aún necesita mejoras. También presenta datos sobre las dificultades que enfrentan las organizaciones para cubrir puestos de trabajo clave y sobre los cambios requeridos en las habilidades laborales. Argumenta que se necesita un cambio en los métodos de enseñanza para enfocarse más en
Este documento ofrece consejos sobre la vida después de la universidad. Primero, resume las opciones disponibles como tomar un año sabático, iniciar un negocio o buscar trabajo. Luego, ofrece recomendaciones para conseguir empleo como completar el servicio social, preparar un CV profesional y practicar para entrevistas. Finalmente, sugiere continuar la educación con una maestría para acceder a mejores puestos de trabajo y mayores ingresos.
Este documento describe cómo crear una Oficina de Dirección de Proyectos (PMO) de manera efectiva. Explica que una PMO debe definir claramente su objetivo, procesos internos, información a administrar, estructura organizacional, y tecnología a utilizar. También debe alinearse con las políticas, cultura y estructura de la empresa, y comunicarse constantemente con su entorno. El documento concluye dando 10 pasos para no crear una PMO de manera efectiva.
La sección "Oferta Académica - Carreras Técnicas" del periódico "El Universal" publicó un artículo el 5 de junio de 2012 sobre las opciones de carreras técnicas y sus perspectivas laborales.
Este documento presenta los objetivos y contenidos de un curso sobre teoría y tendencias actuales de la administración. Los objetivos incluyen explicar los cambios en el pensamiento administrativo, discutir los comportamientos de las empresas, entender los cambios organizacionales necesarios para nuevas tendencias, y analizar estrategias efectivas considerando el entorno. También incluye instrucciones para entregables sobre conceptos de administración, la función administrativa, características de empresas globalizadas y nuevas tendencias organizacionales usando la arquitectura de Zachman.
Este documento ofrece consejos sobre cómo conseguir un empleo y cómo conseguir un mejor empleo después de terminar la universidad. Inicialmente, los recién graduados suelen rechazar ofertas de trabajo por salarios bajos, pero con el tiempo se vuelven más flexibles y están dispuestos a aceptar cualquier trabajo. El documento enfatiza la importancia de crear un currículum vitae profesional, registrarse en sitios de empleo, y prepararse para entrevistas de trabajo mediante la práctica de respuestas a preguntas difíciles. Tamb
El documento habla sobre la situación económica en Chile. Señala que el crecimiento del PIB ha disminuido a menos del 3% y que el desempleo ha aumentado levemente. También menciona que el gobierno ha anunciado nuevas medidas para estimular la economía y crear empleos.
La certificación PMP no garantiza por sí sola que alguien pueda ser director de proyectos. Aunque la certificación demuestra experiencia en gestión de proyectos, el rol de director requiere una amplia gama de conocimientos y habilidades en áreas como análisis de negocios, finanzas, riesgos y liderazgo de equipos que una maestría puede proporcionar pero que la certificación por sí sola no garantiza.
El documento discute la profesionalización de la dirección de proyectos más allá de la certificación, los cursos y la experiencia. Señala que la certificación como PMP no garantiza por sí sola las competencias requeridas para dirigir proyectos de forma exitosa. Una maestría en dirección de proyectos complementa la certificación al profundizar los conocimientos y desarrollar habilidades directivas. La pareja ideal es poseer tanto la certificación PMP como una maestría, lo que brinda un reconocimiento internacional y opciones para asc
El documento describe el valor profesional y organizacional de la dirección de proyectos. Explica que todas las organizaciones necesitan directores de proyectos para realizar proyectos exitosamente en áreas como ingeniería, tecnología, finanzas y mercadeo. Define al director de proyectos como un profesional con experiencia, educación y competencia para liderar proyectos. Describe las actividades clave de un director de proyectos y cuándo se desarrollan durante el ciclo de vida de un proyecto.
Este documento describe la ingeniería social y cómo los hackers la usan para obtener información de seguridad de TI engañando al personal de una organización. Explica que los ingenieros sociales se presentan de manera amigable y hacen preguntas sobre seguridad para obtener información. Recomienda que las organizaciones establezcan políticas y capaciten a su personal para reconocer estas tácticas, y que cuenten con expertos en seguridad de TI que conozcan estas técnicas para prevenir ataques.
Este documento describe cómo resolver sistemas de ecuaciones lineales aplicados a problemas eléctricos usando las leyes de Kirchhoff. Presenta dos ejemplos numéricos que involucran dos y tres ecuaciones lineales respectivamente, mostrando cómo reducir el sistema a uno de menor tamaño y resolverlo para encontrar valores desconocidos como corrientes y voltajes. También propone cuatro problemas adicionales para que los estudiantes practiquen resolviendo sistemas de ecuaciones en contextos eléctricos.
This document provides examples of applying analytic geometry concepts to electrostatics and electricity theory. It introduces the basic equations for lines, circles, parabolas, ellipses, and hyperbolas. For each geometric shape, an example is given of how its equation relates to a concept from electrostatics or electricity, such as Ohm's Law, variation of resistance with temperature, electric power, equipotential curves, and variation of potential with distance. The goal is to help students learn to read technical articles in English and see real-world applications of analytic geometry.
Este documento presenta un plan estratégico de calidad para una organización. Incluye la misión, visión y objetivos de la organización, así como una descripción de sus procesos técnicos y administrativos. Identifica las fortalezas y debilidades de la organización para implementar normas de calidad, y propone normas específicas para cada proceso junto con los pasos, tiempo y responsables para su implantación. El plan busca mejorar los procesos de la organización y su cumplimiento de normas de calidad.
Este documento presenta una introducción al curso "Calidad en la Empresa". El objetivo del curso es que los estudiantes comprendan los principios y métodos de la administración de la calidad total y cómo se han implementado en diferentes organizaciones. El curso cubrirá temas como los enfoques de la administración de la calidad total, técnicas para la planeación y mejora de la calidad, calidad en las relaciones con clientes y proveedores, y la implementación estratégica de la administración de la calidad total.
Este documento presenta una breve introducción a los números complejos y sus aplicaciones en electricidad. Explica que los números complejos surgieron de la necesidad de resolver ecuaciones algebraicas y fueron aceptados lentamente a pesar de su utilidad. Define los números complejos como números de la forma a + bi, donde a es la parte real y b la parte imaginaria. Describe las operaciones básicas entre números complejos como suma, resta, multiplicación y división.
El documento discute la importancia de ver a los empleados como capital humano más que solo recursos humanos. Explica que en entornos turbulentos, las organizaciones deben enfocarse en desarrollar el talento y las habilidades de sus empleados para mantenerse competitivas. También argumenta que la administración del capital humano, en lugar de solo los recursos humanos, es necesaria para traducir la estrategia de una organización en cultura y competencias que crean valor.
Este documento describe diferentes herramientas y técnicas para planear la calidad en un proyecto, incluyendo el costo de la calidad. El costo de la calidad incluye costos de prevención, valoración, fallas internas y externas. Se recomienda implementar un sistema de costo de la calidad, definir métricas de calidad y usar costeo basado en actividades para reducir los costos de la calidad en un proyecto.
Taking AI to the Next Level in Manufacturing.pdfssuserfac0301
Read Taking AI to the Next Level in Manufacturing to gain insights on AI adoption in the manufacturing industry, such as:
1. How quickly AI is being implemented in manufacturing.
2. Which barriers stand in the way of AI adoption.
3. How data quality and governance form the backbone of AI.
4. Organizational processes and structures that may inhibit effective AI adoption.
6. Ideas and approaches to help build your organization's AI strategy.
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
Building Production Ready Search Pipelines with Spark and MilvusZilliz
Spark is the widely used ETL tool for processing, indexing and ingesting data to serving stack for search. Milvus is the production-ready open-source vector database. In this talk we will show how to use Spark to process unstructured data to extract vector representations, and push the vectors to Milvus vector database for search serving.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
Freshworks Rethinks NoSQL for Rapid Scaling & Cost-EfficiencyScyllaDB
Freshworks creates AI-boosted business software that helps employees work more efficiently and effectively. Managing data across multiple RDBMS and NoSQL databases was already a challenge at their current scale. To prepare for 10X growth, they knew it was time to rethink their database strategy. Learn how they architected a solution that would simplify scaling while keeping costs under control.
A Comprehensive Guide to DeFi Development Services in 2024Intelisync
DeFi represents a paradigm shift in the financial industry. Instead of relying on traditional, centralized institutions like banks, DeFi leverages blockchain technology to create a decentralized network of financial services. This means that financial transactions can occur directly between parties, without intermediaries, using smart contracts on platforms like Ethereum.
In 2024, we are witnessing an explosion of new DeFi projects and protocols, each pushing the boundaries of what’s possible in finance.
In summary, DeFi in 2024 is not just a trend; it’s a revolution that democratizes finance, enhances security and transparency, and fosters continuous innovation. As we proceed through this presentation, we'll explore the various components and services of DeFi in detail, shedding light on how they are transforming the financial landscape.
At Intelisync, we specialize in providing comprehensive DeFi development services tailored to meet the unique needs of our clients. From smart contract development to dApp creation and security audits, we ensure that your DeFi project is built with innovation, security, and scalability in mind. Trust Intelisync to guide you through the intricate landscape of decentralized finance and unlock the full potential of blockchain technology.
Ready to take your DeFi project to the next level? Partner with Intelisync for expert DeFi development services today!
leewayhertz.com-AI in predictive maintenance Use cases technologies benefits ...alexjohnson7307
Predictive maintenance is a proactive approach that anticipates equipment failures before they happen. At the forefront of this innovative strategy is Artificial Intelligence (AI), which brings unprecedented precision and efficiency. AI in predictive maintenance is transforming industries by reducing downtime, minimizing costs, and enhancing productivity.
Digital Banking in the Cloud: How Citizens Bank Unlocked Their MainframePrecisely
Inconsistent user experience and siloed data, high costs, and changing customer expectations – Citizens Bank was experiencing these challenges while it was attempting to deliver a superior digital banking experience for its clients. Its core banking applications run on the mainframe and Citizens was using legacy utilities to get the critical mainframe data to feed customer-facing channels, like call centers, web, and mobile. Ultimately, this led to higher operating costs (MIPS), delayed response times, and longer time to market.
Ever-changing customer expectations demand more modern digital experiences, and the bank needed to find a solution that could provide real-time data to its customer channels with low latency and operating costs. Join this session to learn how Citizens is leveraging Precisely to replicate mainframe data to its customer channels and deliver on their “modern digital bank” experiences.
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
GraphRAG for Life Science to increase LLM accuracyTomaz Bratanic
GraphRAG for life science domain, where you retriever information from biomedical knowledge graphs using LLMs to increase the accuracy and performance of generated answers
This presentation provides valuable insights into effective cost-saving techniques on AWS. Learn how to optimize your AWS resources by rightsizing, increasing elasticity, picking the right storage class, and choosing the best pricing model. Additionally, discover essential governance mechanisms to ensure continuous cost efficiency. Whether you are new to AWS or an experienced user, this presentation provides clear and practical tips to help you reduce your cloud costs and get the most out of your budget.
5th LF Energy Power Grid Model Meet-up SlidesDanBrown980551
5th Power Grid Model Meet-up
It is with great pleasure that we extend to you an invitation to the 5th Power Grid Model Meet-up, scheduled for 6th June 2024. This event will adopt a hybrid format, allowing participants to join us either through an online Mircosoft Teams session or in person at TU/e located at Den Dolech 2, Eindhoven, Netherlands. The meet-up will be hosted by Eindhoven University of Technology (TU/e), a research university specializing in engineering science & technology.
Power Grid Model
The global energy transition is placing new and unprecedented demands on Distribution System Operators (DSOs). Alongside upgrades to grid capacity, processes such as digitization, capacity optimization, and congestion management are becoming vital for delivering reliable services.
Power Grid Model is an open source project from Linux Foundation Energy and provides a calculation engine that is increasingly essential for DSOs. It offers a standards-based foundation enabling real-time power systems analysis, simulations of electrical power grids, and sophisticated what-if analysis. In addition, it enables in-depth studies and analysis of the electrical power grid’s behavior and performance. This comprehensive model incorporates essential factors such as power generation capacity, electrical losses, voltage levels, power flows, and system stability.
Power Grid Model is currently being applied in a wide variety of use cases, including grid planning, expansion, reliability, and congestion studies. It can also help in analyzing the impact of renewable energy integration, assessing the effects of disturbances or faults, and developing strategies for grid control and optimization.
What to expect
For the upcoming meetup we are organizing, we have an exciting lineup of activities planned:
-Insightful presentations covering two practical applications of the Power Grid Model.
-An update on the latest advancements in Power Grid -Model technology during the first and second quarters of 2024.
-An interactive brainstorming session to discuss and propose new feature requests.
-An opportunity to connect with fellow Power Grid Model enthusiasts and users.
1. THE ORIGIN AND HISTORY OF CONVOLUTION I:
CONTINUOUS AND DISCRETE
CONVOLUTION OPERATIONS*
ALEJANDRO DOMINGUEZ-TORRES
This work was written while the author was at Applied Mathematics and Computing Group,
Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL, UK. Now the author is at
Academic Division, Fundación Arturo Rosenblueth, México, D.F.1
*This work was sponsored by CONACYT, México. Act Number BA90074.
Chapter 1
Introduction
Nowadays there is not doubt of the uses and applications of the continuous and discrete
convolution operations in many branches of science. Moreover, the number of applications is so
large that trying to name and count them would take a long time. Some of these applications are
in signal and image processing, electric circuits, telecommunications, probability, statistics, etc.
Although the aforementioned applications, all of the modern books dealing with convolution,
which mainly deal with series and integral transforms rather than convolution, do not give any
information about its origin and history. This lack of information has also been present in the
earlier literature. It is worth to mention that the only two books given some remarks about this
history and origin are those by Doetsch [26] and by Gardner and Barnes [37]. The first author
gave some historical remarks in a series of notes marked throughout the text of his book and
commentd at the end of it. In the second book the brief notes of history appear in two subsections
of Appendix C. Both subsections are given in about one page.
From the above comments, it may be seen that the origin and history of convolution have not
been properly traced back. The present paper is an attempt to fill this gap.
Two important remarks must be done about the origin and history of convolution before going on
reading this paper. Firstly, they are not claimed to be complete. The reader may find that the
work of some authors is not fully commented or even mentioned. Secondly, the history given
herein is far away from being a critical one.
This paper has been divided in five chapters plus a list of references. Chapter 2 mainly concerns
with the real convolution integral, while Chapter 3 with the complex convolution integral.
Chapter 4 deals with the discrete convolution operation including some brief comments about
cardinal series. Finally, in Chapter 5 the names and notations given to the different convolution
operations are given.
1 On November 2010, the author is the Corporate Director of Postgraduate Studies at Universidad Tecnológica de
México. alexdfar@yahoo.com
2. 2
In a second paper the author is at present doing an attempt to trace back the origin and history of
the so-called convolution theorems [28].
Chapter2
The Real Convolution Integral
2.1 Definition and Introductory Comments.
Let f,g be two real or complex valued functions of a real variable x. Assume u is a real variable
and construct the integral
b
a
duuxguf )()( , (2.1)
where the limits of integration a and b may finite or infinite or depend on the variable x. These
limits of integration as well as the sign to be taken in the argument of g x u( ) will be apparent in
what follows.
Since both variables x and u entering in (2.1) are real, that integral (2.1) will be called the real
convolution integral (RCI). Obviously these variables are considered as dummy variables and
may be changed through the present chapter.
As in the present, integral (2.1) occurred in the past in several areas of mathematics. Due to this
fact the present chapter has been divided in several sections, in each one of them is traced back
the occurrence of (2.1) in a particular area of mathematics.
It is difficult to establish the exact date when the RCI occurred by the very first time. As it is
described in the following sections of this chapter, the earliest occurrences of (2.1) the present
writer has found are in connection with the theory of series (general, Taylor and trigonometrical
series) and in connection with the theory of Beta function given by Euler.
2.2 Series, Taylor Series and the RCI.
Probably one of the first occurrences of the RCI in an explicit but particular form took place in
the year 1754 when the mathematician Jean-le-Rond D'Alembert (1717-1783) derived Taylor's
expansion theorem on page 50 of Volume I of his book Recherches sur differents points
importants du systeme du monde [73, p.111, footnote 1, and 65, pp.17-18]. Following Reiff
[op.cit., p.111] the derivation of Taylor's expansion theorem by D'Alembert is as follows. For a
function ( )z D'Alembert firstly wrote
( ) ( )z z u ,
Where
3. 3
d
zd
du
)(
. (2.2)
Secondly he wrote
d z
d
d z
dz
v
( ) ( )
,
where now
2
2
)(
d
zd
dv (2.3)
D'Alembert continued with this process and found finally the expression
( ) ( ) ...z z
d
dz
d
dz
1 2
2
2
(2.4)
In this derivation of Taylor's expansion theorem the limits of the integrals (2.2) and (2.3) were
taken from 0 to . Obviously these integrals are special cases of the RCI where one of the
functions entering into them is a constant unit function and the other ones are d dz d dz / , /2 2
,
and so on, respectively.
Reiff [op.cit., p.111] and Nielsen [op.cit., p.18] point out that the series (2.4) was given by
D'Alembert without naming the work of Taylor. Because of this fact, Nielsen [op.cit., p.18] also
points out that Jean-Antoine-Nicolas Caritat de Condorcet (1743-1794) used to designate series
(2.4) as 'théoreme de D'Alembert'.
The above method for deriving Taylor's expansion theorem was known to Pierre Simon Laplace
(1749-1827) since he stated it in 1812 in a more modern form in his book dealing with the theory
of probability [54, pp.179-180]. As in the case of D'Alembert, Laplace did not mention Taylor's
work. Moreover, he did not mention D'Alembert work, the reason of this fact is difficult to guess
since Laplace met D'Alembert when the latter was then at the height of his fame [14, p.259].
In order to see the importance of the RCI in Laplace's method and to distinguish the differences
of it with respect to D'Alembert method, a translation to English language of Laplace's derivation
is given next [op.cit., pp.179-180].
"Consider the integral from z 0 ,
),()()( zxxzxdz
'( )x standing for the differential of ( )x divided by dx. If analogously one designate by
"( )x the differential of '( )x divided by dx., by '''( )x the differential of ''( )x
divided by dx and so on, one will obtain
4. 4
),('')(')(' zxzdzxzzxdz
),('''
2
1
)(''
2
1
)('' 22
zxdzzxzzxdz
Continuing in this way, one will obtain generally
)(
321
)(''
21
)(')(' )(
2
zx
n
z
zx
z
zxzzxdz n
n
)(
321
1 )1(
zxdzz
n
nn
A comparison of this expression with the precedent one, one will have
( ) ( ) '( ) "( ) ( )( )
x x z z x z
z
x z
z
n
x z
n
n
2
1 2 1 2 3
z)(xdzfz
n321
1 1)(nn
.
Setting x z t , the precedent equation will have the following form
( ) ( ) '( ) "( ) ( )( )
t z t z t
z
t
z
n
t
n
n
2
1 2 1 2 3
)z'z(tdzfz
n321
1 1)(nn
,
the integral being taken from z' 0 to z' z ".
Finally it is mentioned that according to Burkhardt [12, p.400, footnote 2065] an expression of
the type
,)()( duuxguf
where the limits of integration are from u 0 to u x ,was used by Sylvestre Françoise Lacroix
(1765-1843) on page 505 of Volume I of his book entitled Traité des différences et des séries.
From the title of Lacroix's book and from the Burkhardt's reference it may be inferred that the
above integral occurred in connection with some work related to series, however no description
of Lacroix's work is given herein since the present writer was not able to obtain Lacroix's book.
5. 5
2.3. Fourier Series and the RCI.
In dealing with the representation of a function by Fourier series several authors used expressions
of the type given by Eq.(2.1). Jean Baptiste Joseph Fourier (1768-1830) himself was the first one
in using such expressions as early as the year 1807 when he made the first announcement of his
investigations about the propagation of heat in solids before the French Academy. In order to
quote the RCI used by Fourier an English translation of Fourier 1822 book is used herein as main
source of reference [36].
In Art.235 of Fourier's book is found the following representation for a function F x( ) defined in
an interval from to (the notation is that used in the translation of Fourier's book)
)(2cos)cos(
2
1
)(
1
)(
xxdFxF ,
which was reduced by him to one of the form
)(cos
2
1
)(
1
)(
xidFxF , (2.5)
where the summation was taken from i 1 to i and where the limits of integration were set
to be from to . There is not doubt that (2.5) is a particular case of the RCI.
In Arts.415-416 of his book Fourier generalized the series to the case of integrals and from the
expression
0
px)dpcos(paf(a)da
p
1
f(x)
he derived the following RCI representation for f x( )
xa
px)sin(pa
f(a)da
p
1
f(x)
when p .
In 1815 Siméon Denis Poisson in a paper submitted to the French Academy at the end of 1815
and in order to participate for the 'Prix d'analyse mathématique' derived, independently of
Fourier, the representation of a function by Fourier series [68]. In pages 85-86 of Poisson's work
it is found the following key RCI-expression for the derivation of such representation (the
notation differs slightly from that used by Poisson)
6. 6
f(a)da
2l
1
l
da
l
a)ip(x
ki)cosexp(
i
l
p
a)(x2cosexp(k)2l
f(a)dak)exp(exp(k)
where the integrals were taken from 0 to 1. Some variations of the above RCI were used
by Poisson in future works [69,70].
Peter Gustav Lejeune Dirichlet (1805-1859) in dealing with the same problem of Fourier series
wrote in 1829 [22] that that representation could be written in the (RCI) form (nowadays known
as Dirichlet integral)
d
xn
x
)(sin2
))((sin
)(
1
2
2
1
.
Georg Friedrich Bernhard Riemann (1826-1866) in 1866 in his work related to Fourier series also
used RCIs to prove some properties of the Fourier coefficients [74]. Among these properties was
included the so-called Riemann-Lebesgue Lemma. A description of his work can be found on
pages 244-247 in the book by Umberto Bottazinni [8].
Finally it is mentioned a result closely related to Fourier series obtained in 1885 by Karl
Weierstrass (1815-1897). Weierstrass proved that [98] if f x( ) is a single-valued function which
is continuous in an interval ( , )c c , then at any point x inside the interval
,,)(lim)( dtn
c
tx
tfxf
c
c
n
where ( , )v n denotes
n
r
r
rvn
1
cos21
2
and n stands for
(n 1) ( m/( n 1))2
with m 1.
2.4 Differential Equations and the RCI.
In the field of differential equations (DE) some authors; v.g. [49, pp.191-192], [71, pp.104-109];
without given their source of reference attribute to Leonhard Euler (1707-1783) the use of a RCI
of the type
7. 7
duuvuxxy v
)()()( 1
(2.6)
for solving "any linear differential equation in which the coefficient of y r( )
is a polynomial in x
of degree r " [71, p.104]. A more inaccurate quotation is given in the book by Gardner and
Barnes [37, p.364]; these authors did not mention neither the exact RCI used by Euler, nor the
type of linear DE that is solved by such expression and nor their source of reference.
On the other hand, the present writer found that the problem of solution of DE by definite
integrals was studied by Euler in 1768 in Volume II of his Institutionum Calculi Integralis [32,
Vol.III], and to be more specific in Chapter X (Caput X: Constructione Aequationum Differentio-
Differentialium per Quadraturas Curvarum).
In problem 131 (Arts. 1049-1052) of that chapter it is found an study of the solution of the DE
L u
d y u
du
M u
dy u
du
N u y u( )
( )
( )
( )
( ) ( )
2
2
0
by means of the integral
b
a
n
dxxQuKxPy .)()()( (2.7)
Obviously (2.7) reduces to (2.6) in the very especial case in which K u u( ) and Q x x( ) .
Euler did not studied explicitly this special case, however he did study only the case Q x x( )
[op.cit., Arts.1050-1052].
At this point the following question arises, Is the preceding citation the source of reference of the
aforementioned authors? If the answer to this question is affirmative, then the facts have been
distorted in time. If the answer is negative, then that source of reference remains as a mystery to
be solved.
In the XIX century the solution of some DE, mainly partial DE describing physical phenomena
were expressed by means for RCIs. One of the earliest author in doing this was Fourier. In fact,
in Chapter IV of his book [36] it is found that the solution of the heat equation
t
k
x
x f x
2
2
0; ( , ) ( ) (2.8)
for a ring was expressed as
2
0
2
).(exp)(cos)(
2
1
),(
i
i
ktixifdtx
8. 8
Fourier went a step further and changing ( , )x t by v x t( , ) and f x( ) by ( )x in (2.8), he solve
the resulting equation for "the free movement of heat in an infinite line" and found for v the
expression
).2()(exp
1 2
ktqxqdqv
(2.9)
It is of historical interest to point out that according to Grattan-Guinness [40, pp.41-42] the
solution of (2.9) was first stated by Laplace in 1809 [53, pp.235-244] and then considered by
Fourier in his future research.
Fourier expressed also (2.9) in a slightly different form for the case in which k 1. He considered
the change of variable q x t ( ) / 2 and expressed (2.9) as
.
p2
/4td)-(x-(a)exp
dav
2
(2.10)
The 3D version of (2.10) was also given by Fourier in connection with the solution of the "free
movement of heat in and infinite solid". Thus in Art.376 of his book it is found the expression.
4t
g)(zb)(ya)(x
g)expb,f(a,dgdbdatv
222
3/2
In Art.384 of the same book the corresponding expression for the case k 1 was also given.
These are probably the first time that a 3D RCI was stated explicitly.
At this point it is worth to mention that Weierstrass in 1885 [98] proved his famous theorem of
approximation of continuous functions by polynomials using an integral of the type given by
(2.10).
One of the first treatises for solving DE by means of RCIs is due to H. Mellin. Indeed, in 1896
Mellin published a paper [63] dealing with the relation of the RCI integral
)(
;)()(
l
dtttx
where (l ) denoted either one of the intervals ( , ),( , ), ( , )a b a b x a x b x or being a b, constants;
to the DE
( ) ( ) ( ) .a b x
d y
dx
a b x
d y
dx
a b x yn n
n
n n n
n
n
1 1
1
1 0 0 0
In Arts.4-5 of his paper Mellin stated the formulae
9. 9
b
a at
b
a
y
x
f,
dt
d
Ft)f(t)dty(x
x
d
Ff(t)
dt
d
Ft)dty(x
,
xb
a
xb
a at
y
x
f
dt
d
Ft)f(t)dty(x
dx
d
Ff(t)
t
d
Ft)dty(x
,
xb
xa
xb
xa
t)f(t)dty(x
dx
d
Ff(t)
dt
d
Ft)dty(x ,
b
a
b
a
t)f(t)dty(x
dx
d
Ft)y(x
x
Fdtf(t)
,
xbt
xb
a
xb
a
y
x
f,
dt
d
Ft)f(t)dty(x
dx
d
Ft)y(x
x
Fdtf(t)
,
xbt
xat
xb
xa
xb
xa
y
x
f,
dt
d
Ft)f(t)dty(x
dx
d
Ft)y(x
x
Fdtf(t)
,
where
F ck
k
k
n
( )
0
,
F
F F
ck
k
n
k v
v
k
v
,
( ) ( )
0
1
0
1
,
being the n.,,0,1,k,ck constants.
it is worth to mention that in Art.8 Mellin made an study of the RCI attributed to Euler:
(l)
a
f(t)dt;t)(x
and in Arts. 13-14 he used the above formulae in conjunction with Convolution Theorem for
Laplace integrals to state existence theorems for the solution of the aforementioned DE.
In pass it is mentioned that in this same paper Mellin introduced the so-called Mellin convolution
and developed similar results than those given for the RCI.
2.5 Integral Equations and the RCI.
In 1823 and 1826 Niels Henrik Abel (1802-1829) published two papers [1, 2] which are rightly
believed to be the earliest publications to contain what is now called an integral equation. The
second of these papers is a revised an improved version of the first. one. Abel solved in these
papers the famous problem of tautochrone curves by reducing it to an integral equation which
now bears his name. To be specific, Abel in the second paper considered the following situations:
10. 10
"Let BDMA be a curve whatever. Let BC be horizontal straight line and CA be a vertical
straight line. Suppone that a particle urged by gravity moves on the curve, a point D whatever
being its point of departure. Let be the time which passed when the particle is at the given
point A, and let a be the height EA [DE and MP are horizontal; E and P are on CA].
The quantity shall be a certain function of a which shall depend on the form of the curve.
Reciprocally, the form of the curve shall depend on this function. We proceed to examine how
with the help of a definite integral, one can find the equation of the curve for which is a
given continuous function of a".
Letting AM s and AP x and t be the time taken by a particle in running through the arc
DM , then if ( )a it follows that
a
xa
ds
a
0
)( .
Abel solved this equation for the variable s and obtained the expression
x
ax
daa
s
0
)(1
. (2.11)
Moreover Abel solved for s the more general expression
a
n
xa
ds
a
0
)(
)( (2.12)
and obtained
0
1
)(
)(sin
n
ax
daan
s . (2.13)
Abel's equation and several others analogous to it were solved by Joseph Liouville (1809-1882)
using the notion of fractional derivatives and integrals. Liouville's procedure was purely formal
and he seemed to be unaware of Abel's work. The solution of (2.12) published by Liouville in
1832 [57, 58] is easily derived from (2.13). Indeed, if it set
x
dvxs
0
,)()(
where v( ) is a function such that v( )0 0 , then a substitution in (2.13) gives
11. 11
x
n
x
ax
daan
dv
0
1
0
,
)(
)(sin
)(
or equivalently
x
n
ax
daa
dx
dn
xv
0
1
)(
)(sin
)(
. (2.14)
At this point it is worth to mention that at the end of the XIX century integral equations of the
RCI type were considered in detail by Volterra. an brief account of Volterra's work is given
below in Section 2.6
Returningto the works by Abel and Liouville, the equations derived by them were the source of
inspiration of several authors to derive and define fractional derivatives and integrals. Complete
accounts of this subject are given in Chapter I of the book by H.T. Davis [20] an in two papers by
B. Ross [75, 76]. Herein only the fractional operations of the RCI type are briefly quoted next.
In the field of fractional calculus, Riemann in 1847 was the first author after Abel and Liouville
in using an integral of the type (2.12). He sought a generalization on Taylor's series expansion
and derived the following expression for fractional integration [75, p.6 and 76, p.4]
x
c
r
r
r
dkkurx
rdx
xud
.)()(
)(
1)( 1
(2.15)
In a paper written in 1863 and published in 1864, H. Holmgren [20, p.20 and 76, p.7] considered
Riemann's expression (2.15) as his point of departure for a long memoire on the subject, and later
on in 1867 applied his theory to the integration of a differential equation of the type [20, p.20 and
76, p.7]
( ) ( ) .a b x c x
d y
dx
a b x
dy
dx
a y2 2 2
2
2
2 1 1 0 0
It is worth to notice that the method used by Holmgren resembles the supposed (and up to now
not confirmed) method used by Euler in the solution of DE by definite integrals (see Sec. 2.2. of
this chapter).
Similar integral to those of (2.15) were studied later by A.D. Grünwald in 1867 and A. V.
Letnikoff in 1872, both in connection with fractional operations and the solution of particular
integrals [76, pp.7-8].
2.6 Volterra's Work and the RCI.
The modern theory of integral equations was initiated almost simultaneously by Eric Ivar
Fredholm (1866-1927) and by Vito Volterra (1860-1940) in the last decade of the XIX century.
12. 12
Volterra's work is based in what he called functions of lines. The theory of these functions of
lines began in 1887 with a series of papers published by Volterra in the Rendiconti de la Real
Accademia dei Lincei and which were condensed and resumed by Volterra himself and J. Pérès in
two books [95, 96], from which the following quotations are taken.
In order to understand how the RCI emerged from Volterra's work, some preliminary definitions
are firstly given [96, pp.5-6].
Given two functions f x y( , ) and g x y( , ) of two variables, the integral (Volterra's notation)
y
x
y)(x,gfy)dxx)g(x,f(x,
was called by Volterra composition of the first kind, while the integral
b
a
y),(x,gfy)dxx)g(x,f(x,
where a band are constants, was called by him composition of the second kind.
Volterra also mentioned that [95, p.100]:
"the operations of composition are an extension to the case of an infinite number of variables
of the notion of the product of two square matrices a bir rsand ; or the composition of the
corresponding linear substitutions ( , , , , ,..., ).i r s n 1 2 Composition of the second kind
correspond to the case of general square matrices, while that of the first kind correspond to the
case of matrices a a r iir irin which for 0 "
Corresponding to these type of composition, Volterra defined two different types of
permutability. The permutability of the first kind was defined as
y
x
x)dxx)g(x,f(x,y)(x,gf
y)(x,fgx)dxx)f(x,g(x,
y
x
while the permutability of the second kind was defined a similar way where the integrals ran from
a bto .
A name for f g
was also given by Volterra [96, p.6]:
13. 13
"Nous dirons que cette fonction f g
est la resultante de la composition de f get ".
The name resultante was later used by several authors to designate the RCI (see Sec. 5.3. herein).
Volterra also considered the very special case in which the function f x y( , ) is permutable with a
constant, say the unity [96, p.9 and 95, pp.109-110]; i.e.,
y
x
y
x
dyfdxf .),(),(
From this equality was where Volterra derived the RCI. His idea is a follows, let ( x,y )be the
common value of these two quantities. By differentiating it is obtained
f ( x,y )
y x
or equivalently
x y
0 .
This partial differential equation implies that ( x,y )and therefore f x y( , ) functions only of the
difference y x . Further, it can be shown that all functions of the difference y x are permutable
(of the first kind) with another and with the unity. In fact, let x y , then
y
x
y
x
;)dx)f(yg()dx)g(yf(
i.e;
f g( y x ) g f ( y x ).
The set of functions permutable (of the first kind) with the unity was called by Volterra group of
the closed cycle in connection with its applications to the theory of heredity.
The group of the closed cycle coincide with the set of functions of one variable y x t . Indeed,
the composition of two functions f gand belonging to the group can be written as
14. 14
t
0
y
x
t
0
(t)gf)d)g(f(t
)d)g(tf()dxx)g(yf(
He also noticed that f g(t )
is a new function of the group. This is the way Volterra arrived to the
RCI.
It is not difficult to see at this point that the so-called group of the closed cycle coincides with
what nowadays is known as linear-shift(-time) invariant (LSI) systems.
In order to illustrate the theory developed by Volterra the following example given by him is
considered [95, Chap.VI, Sec.4]. If represents the angle of torsion and P the torsion couple,
the relation between them is to a first approximation
kP,
where k is a constant determined from physical considerations. but actually the relationship is
more complicated than this since depends not only upon P but also upon the history of the
elastic body, the torsion of which is being studied. This second approximation is the form of an
integral equation
t
dssPstKtkP ,)()()( (2.16)
where K t s( ) is the coefficient of heredity.
Equations of the type (2.16), where the RCI is a characteristic feature of it, are instances of a
general theorem characterizing the so called LSI systems. The formulation and some history of it
are the purposes of the next section of this chapter.
2.7. LSI Systems and the RCI.
Consider a LSI system whose physical action is completely described mathematically by a linear
operator A . Let u x( ) be the function defined by the expression
0if0
0if1
)(
x
x
xu
and let c x( ) be the response of A to function u x( ); i.e., c x A u x( ) ( ) . For an arbitrary function
f x x h x A f x( ), , ( ) ( ). 0 let Then, under the above conditions, it is possible to express h x( ) in
terms of f x c x( ) ( )and . This expression whose derivation can be found in [37] is given as
15. 15
x
0
0.x,)d(x)c(xf'f(0)c(x)h(x) (2.17)
Equation (2.17) was derived and used by Jean Marie Constant Duhamel (1797-1872) in at least
two of his papers [30, 31]. Some authors; e.g., John R. Carson attaches the name of Duhamel to
this equation [17, p.16, footnote 1], however this does not seem to be justified since Liouvill's
solution of Abel's equation [see Eq.(2.14)] is of this form and occurred one year before of
Duhamel's papers.. Indeed, (2.17) can be expressed as
x
0
,.)d)c(xf(
dx
d
h(x)
which is exactly of the form of Eq.(2.14).
Independently of the above derivations, L. Boltzman in 1874 and J. Hopkinson in 1877 obtained
similar expressions to that of (2.17) in the solution of some problems of physics [5 and 48]. Some
authors also attached the name of these authors to that equation [13, p.56].
In the present century, Eq.(2.17) was independently derived and published by Carson [17, p.16,
footnote 1]. The exact reference to his publication is not given in his book (it is the guess of the
present writer that the publication was around the years 1917-1919). As it was pointed out before,
Carson credited Duhamel the derivation of (2.17). He also mentioned that (2.17) was
independently communicated to him by H.W. Nichols and by Stuart Ballantine. However the
exact references were not given.
2.8 Special Functions and the RCI.
There is a great amount of occurrences of the RCI in the theory of special functions. These
occurrences appeared mainly in the formulation of some of their properties and it seems that they
were not formulated by the respective authors having the RCI in mind. In this final section of this
chapter some of those occurrences are quoted chronologically without discussing their derivation.
Euler was probably the first author in using the RCI in the theory of special functions when he
used this integral in his discussion of the Beta function (the Greek letter B was first introduced by
Jaques P.M. Binet in 1839 [15, Vol.II, p.272] ). Indeed in 1768 Euler wrote the expression [32,
Vol,I, Chap.XII, pp.269-270]
,
z)(u
dzz
u
1
1
3
3
2
2
1
1
v
1v
m
(2.18)
where the limits of the integral are to be understool from z z 0 to . Note that the
transformation x z u z / ( ) gives
16. 16
0
1
0
11
1
;)1(
1
)(
dxxx
uzu
dzz v
v
v
an expression which was also proved by Euler.
In the year 1848 Oskar Schlömilch (1823-1901) applied the Gamma function to solve some
definite integrals and stated the following expression [79, Erste Abtheilung, pp.110-11]
x
0
x
0
x
0
131m1l
)d()(x...d)(d)(x
x
0
13ml
)dr.F()(x
s)m(l
(s)(m)(l)
Schlömilch also made reference to the formulas found by Abel [see Eqs.(2.11) and (2.13)].
Concerning Legendre polynomials, in 1848 F. Neumann found that the Legendre polynomials of
the second kind Q zn ( ) could be expressed in terms of the Legendre polynomials of the first kind
P zn ( ). The expression he found was [106, p.320]
1
1
,
)(
2
1
)(
yz
dyyP
zQ n
n
where n is a positive integer and z is a real number between -1 and 1. A second property of the
RCI type for the Legendre polynomials is that given by H.V. Lowry in 1932. This property is
[71, pp.31-32]
x n
n
n
n
x
tdttxP
0
2
12
2 .
2
sin
sin)cos(
Finally some properties of the RCI type concerning Bessel and associated functions are quoted.
These are
0
00 ;)(
)(sin2
)( dxxJ
kx
kx
kJ
0
00 ;)(
)cos(2
)( dxxJ
kx
kx
kY
x
nm
nm dt
t
tJtxJ
nxJ
0
.
)()(
)(
17. 17
The first two formulae were given by N.J. Sonine in 1880 in his study of cylindric functions [89
and 97, p.433]. The last formula was derived in 1905 by H. Bateman using convolution theorem
for Laplace trnasform [97, p.380]. Some special cases of this last formula were derived
independently of Bateman by W. Kapteyn in three papers published in the period 1905-1907.
Bateman in 1812 also used the following integral for some developments of the potential function
[97, p.389]
.)()()(exp 22
0 dttfytxkJkz
Chapter 3
The Complex Convolution Integral
3.1. Definition and Introductory Comments
Let f gand be two complex valued functions of the complex variable z . If is a complex
variable, the integral
C
dgzf
i
)()(
2
1
(3.1)
where i2
1 and C is a suitable curve in the complex plane, will be called the complex
convolution integral (CCI).
The history of the CCI started when the theory of complex integration started to be developed. As
it is well know, the main author who contributed to this development was Augustin-Louis
Cauchy (1789-1857): The brief historical quotations of the CCI given in this chapter start with
the work of this author.
3.2. Cauchy's Integral Formula and the CCI.
Following Bottazinni [8], a particular form of (3.1) was formulated by Cauchy in 1830 in a long
article presented to the Academy of Science of Turin in 1831. Cauchy also gave several
reformulations of this particular form in two papers appeared in 1834 and 1841. These three
papers have an interesting history which is also given in Bottazinni's book [op.cit., pp.157-158].
The aforementioned formulation of Cauchy is not other than the formulation of the so-called
Cauchy integral formula in the theory of functions of complex variable. If f is a continuous and
finite function for x X together with its derivative and where x X ip exp( ) for p ,
then according to Bottazinni [op.cit. p.158], Cauchy stated the formula
.
)(
2
1
)( dp
xx
xfx
xf (3.2)
18. 18
This formula seems to be a RCI at first sight. However after performing some algebraic
manipulations into it, it can be seen that it is a CCI. Indeed, the right hand side of (3.2) can be
written as [8, pp.176-177]
C
p
p
p
p
.xd
xx
)xf(
2pi
1
exp(ip)dpiX
xx
)xf(
2pi
1
dp
xx
)xf(x
2p
1
(3.3)
Obviously (3.3) is a CCI and is the way Cauchy integral formula is known nowadays.
Eq.(3.2) was used by several authors as soon as Cauchy papers were published. The next
important application of this formula was given by P.A. Laurent (1813-1854) when he
established his famous series expansion theorem in 1843 [56]. The CCI appeared in Laurent's
paper when he stated that the coefficients of the series expansion.
f x a a x s a x s a x sn
n
( ) ( ) ( ) ( ) 0 1 2
2
b
x s
b
x s
b
x s
1 2
2
2
3
( ) ( ) ( )
,
where f is an analytic function regular in the open annulus R x s R ', are given as
' 1
;
)(
)(
2
1
C nn dt
st
tf
i
a
C
1n
n dt;s)f(t)(t
2pi
1
b (3.4)
with C' and Care the circles whose common center is the point s and whose radii are R' and R,
respectively.
Concerning the theory of Legendre polynomials P xn ( ), L. Schläfli in 1881 [78] used the results
of Cauchy and Laurent to show that P xn ( ) admitted the following CCI representation [47, p.30]
C nn
n
n d
xi
xP ,
)(2
)1(
2
1
)( 1
C being a large circle whose center is the point x.
On the other hand, it is not difficult to show that the an 's in (3.4) are equivalent to the expressions
' 1
)(
!
1
)(
)(
2
1
C
n
nn
ds
sfd
n
dt
st
tf
i
a
.
19. 19
Thus if r is allowed to be a non-negative number, then the above expression suggests to define
fractional derivatives of f x( ) by means of the formula
C n
n
x
dtxt
tf
i
n
xfD
,)(
)(
2
!
)( 1
(3.5)
where C is a closed curve in the complex plane about the point t x . Eq (3.5) is also a CCI and
by means of that H. Laurent (not to be confused with P.A. Laurent) in 1884 defined fractional
derivatives [20, p.66].
3.3. Pincherele's Work and the CCI.
Although the aforementioned occurrences and uses of the CCI, none of the above authors made a
complete study of (3.1). The earliest study of that equation is perhaps that made by S. Pincherele
in 1908 [67]. Pincherele's study was made in connection with the solution of the complex integral
equation
Pz
zgdzzfzsk
i
),()()(
2
1
(3.6)
where P 0and k z( ) and g z( ) are given functions while f z( ) is unknown. Pincherele
succeeded in the solution of (3.6) using as tool the unilateral Laplace transform. His results and
the results of other authors are resumed in Chap. 17 of Gustav Doetsch's book [26].
Chapter 4
The Discrete Convolution
4.1. Definition and Introductory Comments.
Let ix , and iy be two real or complex sequences such that i . The discrete
convolution (DC) of these sequences is a new sequence defined by the expression
x y in i n
n
(4.1)
Notice that if a sequence, 1;N,0,1,i,x 1i has a number of terms N1 and the sequence
.,1,,1,0, 2 Niyi has a number of terms N2, then the DC of these sequences may be written
in the form
x y , i 0,1, ,N 1; N N N 1n i n 1 1 2
n 0
i
. (4.2)
20. 20
The DC of two finite sequences having N1 and N2 terms, respectively, therefore is a new
sequence having N N1 2 1 terms.
4.2. Cauchy's Work and the DC
The earliest study of the DC is perhaps that performed by Cauchy in his famous book entitled
Cours D'Analyse de L'École Royale Polytechnique which appeared in 1821 [18]. It is of
importance to point out that Cauchy did not give in any part of his book the source of references
he used to establish his results concerning DC and the other topics studied in it, therefore it is
difficult to say if a previous author studied or made use of the DC.
In what follows in this section the results stated by Cauchy are quoted. These quotations are
taken from the aforementioned book by Cauchy.
The first result concerning DC is stated in Chap.IV in connection with the multiplication of
series. On page 141 is established and proved that if
u u u un
n
0 1 2
0 1 2
, , , , , ;
, , , , ,
(4.3)
are two [absolutely] convergent sequences composed only of positive terms and having sum s
and s', respectively, then
u v
u v u v
u v u v u v
u v u v u v u vn n n n
0 0
0 1 1 0
0 2 1 1 2 0
0 1 1 1 1 0
,
,
,
will be a new convergent sequence having sum ss'
The condition of positiveness of the terms in the sequences given in (4.3) was removed by
Cuachy on page 147 and then he proved the corresponding results for real arbitrary [absolutely]
convergent sequences.
The preceding two results were then used by Cauchy to establish a theorem and three corollaries
concerning the multiplication of power series. The theorem is given on page 157 and estates that
if the two sequences
a a x a x a x
b b x b x b x
n
n
n
n
0 1 2
2
0 1 2
2
, , , , ,
, , , , ,
(4.4)
21. 21
are convergent for certain value of the variable x and such that they have sums s and s',
respectively, then
a b
a b a b x
a b a b a b x
a b a b a b a b xn n n n
n
0 0
0 1 1 0
0 2 1 1 2 0
2
0 1 1 1 1 0
,
( ) ,
( ) ,
( )
will be a new convergent sequence which have sum ss'.
The first corollary of this theorem is given on pages 157-158 and in it Cauchy stated that under
the conditions given to the sequences (4.4) the product of series in given as
( )( )
( ) ( ) .
a a x a x b b x b x
a b a b a b x a b a b a b x
0 1 2 0 1 2
2
0 0 0 1 1 0 0 2 1 1 2 2
2
and then he concluded saying that the product of the sums of two sequences is a new sequence of
the same form.
In the second corollary [op.cit., p.158] he extended the result of the first corollary to the case
when an arbitrary number of series is taken into account. In the third corollary [op.cit., p.158] he
considered the very special case in which in (4.4) a b0 0 , a b a b1 1 2 2 , , , and he wrote the
expression
( ) ( ( ) ).a a x a x a a a x a a a x0 1 2
2 2
0
2
0 1 0 1 1
2 2
2 2
Following the analysis given on page 159, he replaced in (4.4) the sequence b b x b x0 1 2
2
, , , by a
polynomial composed of a finite number of terms and:
"on obtient une formule qui ne cesse jamais d'etre exacte, tant que la série
a a x a x0 1 2
2
, , ,
demeure convergente".
Under the above discussion he then established that if the sequence a a x a x a xn
n
0 1 2
2
, , , ,
converges, the product of the sum of this sequence by the polynomial
kx lx px qm m
1
,
where m is an integer number, is a new convergent series of the same type where the general
term will be
22. 22
( ) ,qa pa la ka xn n n m n m
m
1 1
"pourva que l'on considère comme nulles dans les premieres termes celle des quantités
a a a an n n m n m 1 2 1, , , ,
qui se trouveront affectées d'indices négatifs: en d'autres termes, on aura
( )( )km lx px q a a x a xn m
1
0 1 2
2
qa qa pa x0 1 0( ) ( )qa pa la ka xm m
m
1 1 0
( )qa pa la ka xn n n m n m
m
1 1 "
On the other hand, Cauchy also considered the case in which the sequences involved in (4.3) take
complex values. The theorem concerning this case is established on page 283 of his book.
Finally, it is of importance to mention that Cauchy also approached the DC from the point of
view of double sequences, which are studied in NOTE VII of his book. On pages 542-543 he
stated that if
u u u u0 1 2 3, , , , ;
0 1 2 3, , , ,...
are two convergent sequences having sums s and s' and if the following table is constructed
u u u u0 0 1 0 2 0 3 0 , , , ,
u u u0 1 1 1 2 1 , , ,
u u0 2 1 2 , ,
u0 3 ,
then the vertical sums
u0 0 ,
u u0 1 1 0 ,
u u u0 2 1 1 2 0
,
u u u un n n n0 1 1 1 1 0
will be a new convergent sequence, and the sum of this new sequence will be equal to ss'.
As it can be seen at this point, the discussion made by Cauchy concern both (4.1) and (4.2).
Improvements of his results concerning convergence were given by several authors in the second
half of the las century. These results can be found, for example, in the book by G.H. Hardy [44].
4.3. Statistics and the DC.
23. 23
An application of discrete convolution, probably the first one and which seems no to be in
connection with Cauchy's work, was given by actuaries and vital staticians in the XIX century.
This application was mainly in dealing with the problem of graduation of statistical data by
linear compounding. The main idea of this method consists of the replacement of a sequence of
observed values ru (of a sequence of true values rU ) by a sequence r , where each vr is
obtained by a linear compound given by the expression
v b u b u b ur r r r r r r ( )1 1 1 1
( )b u b ur r r r2 2 2 2
(4.5)
( )b u b ur n r n r n r n
for a range of 2 1n terms and on the assumptions that the finite differences of rU beyond
certain order j may be neglected.
Assume now that the b s' in (4.5) are such that b b ar k r k k , then (4.5) becomes
r r r r r r n r n r na u a u u a u u a u u 0 1 1 1 2 2 2( ) ( ) ( ).
It is not difficult to see that this expression is a DC formula. Indeed, it may be written as
r k r n
k n
n
a u
(4.6)
The fundamental conditions under which such replacements of ur by linear compounding of the
u s' are legitimate were well set out by W.F. Sheppard in three papers published in the period
1912-1915 [86, 87 and 88]. See Hugh H. Wolfenden's paper for an account of these papers [104].
The determination of the b s' or a s' may be affected by interpolation, fitting by least squares, or
simply reduction of error processes. According to Wolfenden [op.cit., p.83], the earliest
application of (4.6) in interpolation was that of Griffith Davies in 1834 in connection with the
mortality table of the Equitable Society. The application of (4.6) to the problem of fitting by least
squares were indicated briefly by C.L. Landré in 1901 [52], and were fully worked out by
Sheppard in his papers. The formulae for reduction of error were indicated by G.F. Hardy (not to
be confused with G.H. Hardy) in 1909 [43] and fully examined by Sheppard in his papers, and
some of them afterward were rediscovered independently by R. Henderson in 1916 [45] and J.
Larus in 1918 [55]. An account of these papers may be found in Wolfenden's paper and book
[104, 105].
The paper published by Wolfenden in 1925 was based on the work by, until then unknown,
Erastus L. De Forest and then it became known that the determination of the a s' in (4.6) in the
case of interpolation, fitting by least squares and those of reduction of error, which make the
mean square error in 4
a minimum, had previously been discussed very fully by De Forest.
These discussions appeared in a series of papers published in the Smith sonian Reports of 1871
24. 24
and 1873, in a pamphlet in 1876 and in 1877-1880 in a Journal of Des Moines, Iowa, called The
Analyst (A monthly Journal of Pure and Applied Mathematics). The complete list of the papers
written by De Forest is given at the end of Wolfenden's paper.
According to Wolfenden, De Forest also made an extensive investigation of the effects of
applying some of his linear compounding formulae repeatedly, and in this instance also reached
some important conclusions on a matter which was suggested, but not closely examined, by other
authors in later years. De Forest noted clearly the manner in which when a linear compounding
formula is repeated in a large number of times, the curve of the coefficients ultimately tends to a
central bell-shaped potion with an infinite number of small oscillations at each end. To be
specific, De Forest observed that the limiting form of the curve of coefficients of (4.6) becomes
the normal probability curve when these coefficients are symmetric, while in the unsymmetrical
case [see Eq.(4.5)] he reached an unsymmetrical probability curve. It is important to point out
that these observations in the symmetrical case were proved many years later, to be specific, in
1948, by Isaac J. Schoenberg [81].
The next importance occurrence of (4.6) in statistics was in the year 1946 in a paper written by
Schoenberg [80]. This occurrence of (4.6) is closely related with the work of De Forest just
described. Indeed, in that paper Schoenberg approached the problem of smoothing the sequence
of equidistant data ny by a series of the type
F y L , L L ,n n n n n n
n
(4.7)
from the point of view of Fourier series of the functions T u u( ) ( )and whose Fourier
coefficients are nn Ly and , respectively [80, pp.50-56]. In his paper, he characterized the
smoothing properties of (4.7) in terms of ( )u and established the conditions oin ( )u such that
(4.7) reproduces the values of yn of a polynomial of degree not exceeding a given integral
number m. Two years later, Schoenberg returned to the problem of smoothing and, as it was
pointed out before, he proved the De Forest's observation about the bell shaped form of the
iterates of (4.7). In the following years, Schoenberg and his students made several investigations
of the subject of smoothing data by (4.7). These investigations were resumed by him in a paper
published in 1953 [82].
In pass it is worth to mention that in Schoenberg's paper of 1946 the B-splines were introduced
by the very first time in the modern mathematical literature.
4.4. Special Functions and the DC.
In the theory of special functions, and mainly in the theory of Bessel functions, there exist several
properties of them which are given by means of DC formulae. Some of these formulae are briefly
quoted in this section.
As usual let J xn ( ) denote the Bessel function of the first kind and of order n. The earliest
property of J xn ( ) given by a DC formula seems to be given by P.A. Hansen in a paper originally
25. 25
published in German in 1843 and which its translation to French appeared in 1845. The
expression given by Hansen was [42 and 97, pp.30-31]
J x J x J x J x J xr r r r
rr
1 1 1
10
1
2 2( ) ( ) ( ) ( ) ( ).
This expression is a particular case of a more general one derived independently by C.G.
Neumann in 1867 [64, p.40, and 97, p.30] and E.C.J. von Lommel in 1868 [59, pp.26-27, and 97,
p.30]. The general expression is
J y z J y J zn m n m
m
( ) ( ) ( )
(4.8)
This expression for z y and for all n was also derived by L. Schläfli in 1871 [78, pp.135-137,
and 97, p.30].
In 1867 Neumann also derived the following formulae for the case 0 [64, and 97, p.30],
where Y x ( ) denotes the Bessel function of the second kind and order ,
Y Y Z J z mm m
m
( )cos( ) ( ) ( )cos( ).
Y Y Z J z mm m
m
( ) ( ) ( ) ( ) ( )sin sin
,
with Z z Zz2 2
2 cos ; while Lommel derived the expression
( ) ( ) ( ) ( ) ( ) ,
1 2 02
1
2
0
2
r
r n r r
r
n r
r
n
J z J z J z J z
and the espression
Y z t Y t J zm m
m
( ) ( ) ( ),
(4.9)
for z t .
Concerning Schaläfli, he also gave the following expression in his paper of 1871 [op.cit., pp.139-
141, and 97, p.289]
S t z S J z z tn n m m
m
( ) ( ); ;
where S tn ( ) are the nowadays so-called Schläfli polynomials defined by the expression
26. 26
)2/(
0
2
0
.1;
2!
)!1(
)(
,0)(
n
m
mn
n n
t
m
mn
tS
tS
On the other hand, in 1872 L. Gegenbauer in his studies dealing with Bessel functions defined the
so called Neumann's polynomials of order n the formula [38, and 97, p.273 and p.290]
O t
n
n
n m m n
n m t m
m
n
( )
( )cos ( )
( )( )
,
1
4 1
2
2
2
2 2
1
0
In a paper dated August 1879 and published in 1880 N.J. Sonine proved that if C z ( ) is defined
by the expression [89]
C z J z Y z ( ) ( ) ( ) ( ) ( ), 1 2
where 1 2( ) ( )and are arbitrary periodic functions of with period unity, then
C z t C t J zm m
m
( ) ( ) ( ).
(4.10)
Formulae (4.8), (4.9) and (4.10) were also proved by J.H. Graf in a paper dated March 1893 and
published the same year [39, pp.141-142]. In this same paper Graf also proved that [op,cit.,
pp.142-144]
m
m imzJ
izZ
izZ
J ),(exp)(
)(exp
)(exp
)(
2/
where Z z 2Zzcos and zexp( i ) Z .2 2
Finally, outside of the theory of Bessel functions, the following expression was derived by C.
Runge in 1914 [77]
n
r
rnrn
n
yHxH
r
nyx
H
0
/2
),()(
2
2
where H xn ( ) denotes the Hermite polynomials of degree n 0 1 2, , , .
4.5. Cardinal Series.
A convolution formula closely related to the DC is that given by the expression
27. 27
a f z nn
n
( ) (4.11)
Obviously this formula reduces to the DC, depending of the values of n, if z is allowed to take
integral values only. In the literature (4.11) is better known as cardinal series. A major factor
affecting current interest in the cardinal series is its importance in the sampling theory of band-
limited functions or signals. Althouhg this application, its origin concerns with the problem of
interpolation as will be seen bellow. Herein only some historical remarks concerning (4.11) will
be given. For major accounts the reader is referred to the works of A.J. Jerri [50] and J.R.
Higgins [46].
The first explicit use of (4.11) occurs in a brief note by Félix-Edouard-Justin-Emile Borel (1871-
1956) in 1898 in a paper dealing with the problem of expansion of a function by Taylor series
[6]. On page 1002 of this paper it is found a expression of the type
( z )
sin( z ) a
z n
n
n 0
(4.12)
to get information about how the power series coefficients na of a function f x a zn
n
( )
determine its singularities.
The next year, Borel in dealing with the problem of interpolation used the following formula [7]
sin( t ) c ( 1)
t n
,n
n
n
(4.13)
which can be written as
c
t n
t n
n
n
sin
( )
( )
.
On page 83 of this last paper Borel mentioned that he deduced the series from Lagrage's
interpolation formula.
Independently of Borel, in 1900 John Dougall expanded the solution P z ( ) of Legendre's
equation
(1 z )
d y
dz
2z
dy
dz
n(n 1)y 0, z 1,2
2
2
as a series of Legendre polynomials [29]
).()1(
1
11sin
)(
0
zP
nn
zP n
n
n
28. 28
The corresponding expansion for the second solution Q z ( ) of Legendre's differential equation
was given by H.B.C. Darling in 1923 [21].
On the other hand, since P z P zn n 1( ) ( ), it is readily seen that if f P z( ) ( ) then Dougall's
formula may be regarded as a case of the following interpolation formula
f
n
n
f n
n
( )
( )
( )
( )
sin
(4.14)
which was established by Jaques Hadamard (1865-1963) in 1901 [41] after an extensive study of
Borel's paper of 1898.
Hadamard's formula (4.14) is analogous to the fundamental interpolation formula of Charles de la
Vallée Poussin (1866-1962) appeared in 1908. The interpolation scheme due to de la Vallée
Poussin considers the finite interpolation formula [72, p.327]
sinmt
m
f n m
t
n
n
ma
b
( ) ( / )
,
1
where f x( ) is a given function defined on the finite interval a b, , and the summation is
understood to be over those n for which n m a b/ , .
According to W.L. Ferrar [34, p.333], F.J.W. Whipple in an unpublished manuscript dated 1910
introduced the cardinal series and discovered several of its properties, including the band-limited
nature of its sum.
Later, in 1915, Edmund T. Whittaker rediscovered again the cardinal series in connection with
the problem of interpolation of equidistant data [99]. In his paper Whittaker did not make
references to previous work. The expression used by Whittaker was [op.cit., p.86]
f a n
x a n
x a nn
( )
( ) /
( ) /
,
sin
(4.15)
which reduces to (4.14) if m a n . E.T. Whittaker did not call (4.15) cardinal series, the name
seems to first appeared in the works of Ferrar [34] and J.M. Whittaker (second son of E.T.
Whittaker) [101, 102].
After the above rediscovering of the cardinal series, it came up a period where they were used
and extended (this extension considered more general expressions no necessarily of the
convolution type) to deduce properties of entire functions from their known behaviour at a
sequence of points. Among the authors who made extensions are J.F. Steffensen [90], T.A.
Brown [10, 11], M. Theis [91], K. Ogura [66], W.L. Ferrar [33, 34, 35], T.M. MacRobert [60,
61], I.M. Sheffer [85], E.T. Copson [19] and J.M. Whittaker [101,102]. A brief account of some
of these papers are given in Story One of Higgin's paper [46, pp.53-57].
29. 29
The cardinal series (4.15), appeared also in the russian literature. It was firstly given by V.A.
Kotel'nikov in 1933 in dealing with certain problems of communication [51].
In the american literature, it was mainly C.E. Shannon in 1949 who introduced a series of the
type given in (4.15), this was also in dealing with problems in communication [84]. Although
Shannon results were published in 1949, his paper was apparently written in 1940, however its
contents seem to have been in circulation in the United States by 1948.
The general theory of interpolation formulae of the type (4.11) with f x f x( ) ( ) started in
1946 with I.J. Schoenberg. The results obtained by him, his students and other authors in the
period 1946-1973 were stated in his paper of 1946 and in his monograph of 1973, In pass, it is
pointed out that in his paper of 1946 the B-splines were given a name and were used by the very
first type to solve the problem of interpolation.
The most recent account concerning cardinal series from an introductory point of view is given in
the book by R.J. Marks [62].
Chapter 5
The Notations and Names of Convolution
5.1. Introduction.
Some of the convolution operations defined in the preceding chapters have been denoted and
named in several ways in the literature. As it wil be seen below, the notation used for these
operations has been almost uniform in the literature. Concerning the names, these were usually
given either after the work of some author or after the study of the properties of these operations.
5.2. Notations.
Probably the first expression used to denote the RCI is that given by Volterra. Indeed, as it was
seen in Sec. 2.6., Volterra denoted and defined the so-called composition of the first kind as
y
z
y).(x,gfy)dxx)g(x,f(x,
Although the above notation is quite general, in the particular case that f x y( , ) and g x y( , )
belong to the group of the closed cycle (see Sec. 2.6.) Volterra used the same notation and he
wrote:
t
0
(t).gf)d)g(f(t
This notation can be considered as the most primitive notation of the very well known star-
notation or asterisk-notation:
30. 30
t
tgfdgtf
0
).)(()()( (5.1)
It is the guess of the present writer that the star-notation was firstly used by Gustav Doetsch.
Thus for example, Doetsch used the above notation in a paper published in 1927 [25, p.23].
Other notation were also given by Doetsch. For example on page 161 of [26] is stated that the
right-hand-side of (5.1) may be denoted as
f g
t
0
(5.2)
in order to be distinguished from the RCI
,)d)g(f(t
which he denoted as
f g
. (5.3)
This notation also appeared on page 127 et. seg. of a paper published in 1938 by F. Tricomi [93].
Notations (5.2) and (5.3) seem to be difficult to write and have disappeared from the literature.
Doetsch himself did not use them in his book of 1970 [27].
Concerning the notation for complex convolution, Doetsch gave no notation in his paper of 1927
nor in his book of 1943. However in his book of 1970 he used the notation
f go
to designate that operation. On the other hand, Gardner and Barnes on page 275 of their book
[37] used the notation
to designate the CCI of functions F s F s1 2( ) ( )and .
5.3. Names.
As it was seen in Sec. 2.6., in the theory of integral equations developed by Volterra the notion of
composition of two functions f x y g x y( , ) ( , )and played and important role. Considering only
31. 31
the composition of the first kind, it was also seen that if these functions were also permutable
functions with the unity, then
f ( x,y ) f g( x,y ) g f ( x,y )
f g( y x ) g f ( y x )
f ( y x )
from which easily follows the RCI
t
dtgft
0
.)()()( (5.4)
Thus it can be said that the composition reduces to the operation (5.4).
The name composition is one of the first names attadched to (5.4) and has been used and
preferred frequently, since the times of Volterra's work, in the French literature. In that literature
the name has been modified slightly and some times it appears as produit de composition
(product of composition). The name composition has also been used as an alternative name in
some German literature where the name faltung is preferred; e.g., [4, p.56., and p.285 Remark-
Quotation 40] and [26, p.157]. Mereover, the name composition has been extended to designate
the RCI
.)()( dtgf (5.5)
Another name derived from the work by Volterra and Pérès, although less common, to designate
(5.5), is that of resultant [96, p.6]. This is the name used by E.C. Titchmarsh [92, p.51] and by B.
Van der Pol and J. Bremmer [94]. The name resultant has been also used to designate (5.4).
Indeed, in his book Doetsch [26, p.157] suggested that this name may be used as an alternative
English name to designate that operation; while G.H. Hardy in his book stated [44, p.98,
footnote]:
"The German name equivalent of resultant is faltung".
The name faltung to designate (5.4) seems to be given by the very first time by Doetsch in two
papers appeared in 1923 [23, 24]. To be more specific, (5.4) was called by Doetsch
faltungsintegral [25, p.23].
In the period of time immediately after the occurrence of the aforementioned Doetsch's papers,
the name faltung was the most common to designate (5.4) and/or (5.5). In his book Doetsch [25,
p.157] pointed out that the name faltung was preferred by many American authors. This
affirmation is supported by the following quotation taken from the book by Norbert Wiener [103,
p.45]:
32. 32
"The quantity
dxxygxf )()(2
1
is known as the Faltung of f x( ) and g x( ) (there is not good English word), and the
sequence
nmn ba
as the Faltung of the sequences nn ba and ".
Wiener also used the word faltung to designate (5.5) [103, p.71].
Concerning the CCI, on page 23 of his paper of 1927 Doetsch designated this integral as
Facherintegral. However in his book he designated it as komplexe Faltung [27, p.167].
The name faltung has also been used to designate the discrete convolution operation [103, p.45].
On the other hand, it is difficult to say when the name convolution occurred by the very first time
in the literature. Doetsch himself suggested the name as an English translation of the german
name Faltung [26, p.157.]. On page 228 of the book by Gardner and Barnes it is found the
following quotation [37]:
"The process expressed by the integral [(5.4)] will be called convolution in the real domain, or
real convolution, and the functions [entered into it] will be said to be convolved".
On pages 231-233 of the same book by Gardner an Barnes, it was given what was probably the
first graphical interpretation of the RCI. This interpretation was given by convolving the
functions exp( ) ( ). t t tand exp At the end of this example these authors also pointed out that:
"It can be seen from this example that 'convolution' denotes a mathematical process that can be
interpreted graphically by folding, translating, multiplying, and integrating".
The process of folding in this graphical interpretation agrees with the translation to English of the
German word faltung, which means folding. It also agrees with the definition given on pages
952-953 of the 1978 edition of the Oxford English Dictionary. Indeed, according to this
dictionary, the word convolution concerns with the action of folding, and according to its
etymological roots it concerns with the action of rolling up together.
Gardner and Barnes also designated the CCI as [op.cit., p.275]: "convolution in the complex
domain or more briefly complex convolution".
Nowadays the name convolution has become in common use in the literature to designate either
the RCI, or the CCI, or the DC. In the past other names have been used to designate these
operations or special instances and variations of them. For example, the integral
33. 33
x
xdxcxfxcfxh
0
;0;)()(')()0()( (5.6)
which is equivalent to the integral
x
xdxcxf
dx
d
xh
0
;0;)()()(
some times has been referred as superposition theorem or Duhamel's theorem [16, pp.30-31, and
p.301]. The first name is due to, in the derivation of (5.6), the superposition principle plays an
important role [37, p.234]. On the other hand, Duhamel used the superporsition principle to
derive (5.5), this is the reason of the second name. The names of Boltzman and Hopkinson has
been also attached to (5.5) [13, p.56] (see also Sec. 2.7.).
Although the above names were originally given to (5.6), some authors have used them to
designate in general the RCIs. For example, R.B. Blackman and J.W. Tukey on page 73 of their
book [3] pointed out that:
"Convolution is often called by a variety of names such as Superposition Theorem,
Faltungsintegral, Green's Theorem, Duhamel's Theorem, Borel's Theorem, and Boltzman-
Hip'kinson Theorem".
In this book no references were given to this quotation. The name Borel's Theorem in the above
quotation is not justified since in some early literature the name is given to Convolution Theorem
for Laplace integral [28]. The name Green's Theorem remains as a mystery for the present writer
since the aforementioned authors gave no references.
Ronald N. Bracewell on page 24 of his book [9] stated a similar paragraph to the above of
Blackman and Tukey. He wrote:
"The word 'convolution' is coming into more general use as awareness of its oneness spreads
into various branches of science. The German term Faltung is widely used, as is the term
'composition product', adapted from the French. Terms encountered in special fields include
superposition integral, Duhamel integral, Borel's theorem, (weighted) running mean,
crosscorrelation function, smoothing, blurring, scanning, and smearing".
The reader is referred to Bracewell's book for the justification of the last six names.
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