The document introduces the Fourier transformation and some of its key properties and applications. It defines the Fourier transform and inverse Fourier transform integrals. Some examples of Fourier transforms are given, such as the transform of exp(-a|x|). The properties of Fourier transforms discussed include the Riemann-Lebesgue lemma, convolution theorem, and properties of convolutions. Applications of Fourier transforms include solving linear boundary value problems and differential/integral equations.
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document provides definitions and examples related to Fourier series and Fourier transforms. It defines the Fourier transform and inverse Fourier transform of a function f(x). It gives the Fourier integral representation of a function and provides an example of finding the Fourier integral representation of a rectangle function. It also defines Fourier sine and cosine integrals. Finally, it outlines some properties of Fourier transforms, including the modulation theorem and convolution theorem.
This document provides an overview of Fourier series and Fourier transforms. It begins with examples of applying Fourier series to odd and even functions. It then defines the Fourier transform and provides examples of its application. Key properties of the Fourier transform are outlined, including linearity, symmetry, time shifting, and how differentiation in the time domain relates to multiplication in the frequency domain. Important applications of the Fourier transform mentioned include convolution, deconvolution, sampling of seismic time series, and filtering. The document is a lecture on seismic data processing that introduces Fourier analysis techniques.
This document outlines the key topics in mathematical methods including:
- Matrices and linear systems of equations, eigen values and vectors, and real/complex matrices.
- Fourier series, Fourier transforms, and partial differential equations.
It provides textbooks and references for further study. The unit focuses on Fourier series, covering properties of even and odd functions, Euler's formulae, and half-range expansions. It also introduces Fourier integrals and transforms, discussing cosine, sine, and complex forms.
1 IntroductionThese notes introduces a particular kind of HiAbbyWhyte974
1 Introduction
These notes introduces a particular kind of Hilbert space known as a reproducing kernel Hilbert space (RKHS).
We will establish connections with kernels, defined previously, and show that kernels and RKHSs are in one-to-
one correspondence. This material is largely drawn from Chapter 4 of [1], although some results are presented
in a slightly different way to ease digestion.
2 Reproducing Kernel Hilbert Spaces
Throughout these notes, we use the term “Hilbert function space over X ” to refer to a Hilbert space whose
elements are functions f : X 7→ R.
Definition 1. (Reproducing Kernel) Let F be a Hilbert function space over X . A reproducing kernel of F is a
function k : X × X 7→ R which satisfies the following two properties:
1. k (·,x) ∈F for any x ∈X.
2. (Reproducing property) For any f ∈F and any x ∈X, f (x) = 〈f,k (·,x)〉F.
Theorem 1. Let F be a Hilbert function space over X. The following are equivalent:
1. F has a reproducing kernel.
2. For any x ∈X, the function δx : F 7→ R defined by δx (f) = f (x) is continuous.
Definition 2. Let F be a Hilbert function space over X. We say F is a reproducing kernel Hilbert space
over X if it satisfies the conditions of the previous theorem.
Proof. ( (1) ⇒ (2) ) . Suppose that F has a reproducing kernel, k : X ×X 7→ R. We need to show that if
(fn)
∞
n=1 converges to f ∈F, i.e., ‖fn −f‖F → 0, then for every x ∈X , |δx (fn) −δx (f)| = |fn (x) −f (x)|→
0. Now
δx (f) = f (x) = 〈f,k (·,x)〉F = 〈 lim
n→∞
fn,k (·,x)〉F
(a)
= lim
n→∞
〈fn,k (·,x)〉F = lim
n→∞
fn (x) = lim
n→∞
δx (fn)
Note that the identity “(a)” is implied by the continuity of inner product operator of its first argument.
Before proving the reverse implication, we state a classical result from functional analysis named the Riesz
representation theorem.
Theorem 2 (Riesz representation theorem). Let F be a Hilbert space and L : F 7→ R be a linear functional
on F. Then L is continuous if and only if there exists Φ ∈F such that L (f) = 〈f, Φ〉F for any f ∈F.
1
2
( (2) ⇒ (1) ) . Let x ∈ X . It’s trivial to see that δx is a linear functional, so due to the assumed
continuity of δx, Theorem 2 ensures the existence of a Φx ∈ F such that δx (f) = f (x) = 〈f, Φx〉F. Now
define k (x2,x1) := Φx1 (x2) ∈ R. Note that k (·,x) = Φx ∈F (establishing the first property of reproducing
kernels). Also,
f (x) = 〈f, Φx〉F = 〈f,k (·,x)〉F
which shows that the reproducing property holds.
The next result establishes that kernels and reproducing kernels are the same.
Theorem 3. Let k : X ×X 7→ R. Then k is a kernel if and only if k is a reproducing kernel of some RKHS
F over X.
Proof. (⇐) . We first prove the reverse implication. Let k be the reproducing kernel of F and define Φ :
X 7→F by Φ (x) = k (·,x). Now for any x′ ∈X , consider the function fx′ = k (·,x′). Using the reproducing
property and symmetry of the inner product, we obtain
k (x,x′) = fx′ (x) = 〈fx′,k (·,x)〉F = 〈k (·,x′) ,k (·,x)〉F
= 〈Φ (x′) , Φ (x)〉F = 〈Φ (x) , Φ (x′ ...
1 IntroductionThese notes introduces a particular kind of HiMartineMccracken314
1 Introduction
These notes introduces a particular kind of Hilbert space known as a reproducing kernel Hilbert space (RKHS).
We will establish connections with kernels, defined previously, and show that kernels and RKHSs are in one-to-
one correspondence. This material is largely drawn from Chapter 4 of [1], although some results are presented
in a slightly different way to ease digestion.
2 Reproducing Kernel Hilbert Spaces
Throughout these notes, we use the term “Hilbert function space over X ” to refer to a Hilbert space whose
elements are functions f : X 7→ R.
Definition 1. (Reproducing Kernel) Let F be a Hilbert function space over X . A reproducing kernel of F is a
function k : X × X 7→ R which satisfies the following two properties:
1. k (·,x) ∈F for any x ∈X.
2. (Reproducing property) For any f ∈F and any x ∈X, f (x) = 〈f,k (·,x)〉F.
Theorem 1. Let F be a Hilbert function space over X. The following are equivalent:
1. F has a reproducing kernel.
2. For any x ∈X, the function δx : F 7→ R defined by δx (f) = f (x) is continuous.
Definition 2. Let F be a Hilbert function space over X. We say F is a reproducing kernel Hilbert space
over X if it satisfies the conditions of the previous theorem.
Proof. ( (1) ⇒ (2) ) . Suppose that F has a reproducing kernel, k : X ×X 7→ R. We need to show that if
(fn)
∞
n=1 converges to f ∈F, i.e., ‖fn −f‖F → 0, then for every x ∈X , |δx (fn) −δx (f)| = |fn (x) −f (x)|→
0. Now
δx (f) = f (x) = 〈f,k (·,x)〉F = 〈 lim
n→∞
fn,k (·,x)〉F
(a)
= lim
n→∞
〈fn,k (·,x)〉F = lim
n→∞
fn (x) = lim
n→∞
δx (fn)
Note that the identity “(a)” is implied by the continuity of inner product operator of its first argument.
Before proving the reverse implication, we state a classical result from functional analysis named the Riesz
representation theorem.
Theorem 2 (Riesz representation theorem). Let F be a Hilbert space and L : F 7→ R be a linear functional
on F. Then L is continuous if and only if there exists Φ ∈F such that L (f) = 〈f, Φ〉F for any f ∈F.
1
2
( (2) ⇒ (1) ) . Let x ∈ X . It’s trivial to see that δx is a linear functional, so due to the assumed
continuity of δx, Theorem 2 ensures the existence of a Φx ∈ F such that δx (f) = f (x) = 〈f, Φx〉F. Now
define k (x2,x1) := Φx1 (x2) ∈ R. Note that k (·,x) = Φx ∈F (establishing the first property of reproducing
kernels). Also,
f (x) = 〈f, Φx〉F = 〈f,k (·,x)〉F
which shows that the reproducing property holds.
The next result establishes that kernels and reproducing kernels are the same.
Theorem 3. Let k : X ×X 7→ R. Then k is a kernel if and only if k is a reproducing kernel of some RKHS
F over X.
Proof. (⇐) . We first prove the reverse implication. Let k be the reproducing kernel of F and define Φ :
X 7→F by Φ (x) = k (·,x). Now for any x′ ∈X , consider the function fx′ = k (·,x′). Using the reproducing
property and symmetry of the inner product, we obtain
k (x,x′) = fx′ (x) = 〈fx′,k (·,x)〉F = 〈k (·,x′) ,k (·,x)〉F
= 〈Φ (x′) , Φ (x)〉F = 〈Φ (x) , Φ (x′ ...
The document discusses properties and differentiation of natural exponential functions:
1) The natural exponential function f(x) = ex is its own derivative. Taking the derivative of both sides of ln(ex) = x gives f'(x) = ex.
2) Geometrically, the slope of the graph of f(x) = ex at any point (x, ex) is equal to the y-value ex.
3) Examples are provided to demonstrate differentiating exponential functions, locating relative extrema of exponential functions, and showing that the standard normal probability density function has points of inflection at x = ±1.
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document provides definitions and examples related to Fourier series and Fourier transforms. It defines the Fourier transform and inverse Fourier transform of a function f(x). It gives the Fourier integral representation of a function and provides an example of finding the Fourier integral representation of a rectangle function. It also defines Fourier sine and cosine integrals. Finally, it outlines some properties of Fourier transforms, including the modulation theorem and convolution theorem.
This document provides an overview of Fourier series and Fourier transforms. It begins with examples of applying Fourier series to odd and even functions. It then defines the Fourier transform and provides examples of its application. Key properties of the Fourier transform are outlined, including linearity, symmetry, time shifting, and how differentiation in the time domain relates to multiplication in the frequency domain. Important applications of the Fourier transform mentioned include convolution, deconvolution, sampling of seismic time series, and filtering. The document is a lecture on seismic data processing that introduces Fourier analysis techniques.
This document outlines the key topics in mathematical methods including:
- Matrices and linear systems of equations, eigen values and vectors, and real/complex matrices.
- Fourier series, Fourier transforms, and partial differential equations.
It provides textbooks and references for further study. The unit focuses on Fourier series, covering properties of even and odd functions, Euler's formulae, and half-range expansions. It also introduces Fourier integrals and transforms, discussing cosine, sine, and complex forms.
1 IntroductionThese notes introduces a particular kind of HiAbbyWhyte974
1 Introduction
These notes introduces a particular kind of Hilbert space known as a reproducing kernel Hilbert space (RKHS).
We will establish connections with kernels, defined previously, and show that kernels and RKHSs are in one-to-
one correspondence. This material is largely drawn from Chapter 4 of [1], although some results are presented
in a slightly different way to ease digestion.
2 Reproducing Kernel Hilbert Spaces
Throughout these notes, we use the term “Hilbert function space over X ” to refer to a Hilbert space whose
elements are functions f : X 7→ R.
Definition 1. (Reproducing Kernel) Let F be a Hilbert function space over X . A reproducing kernel of F is a
function k : X × X 7→ R which satisfies the following two properties:
1. k (·,x) ∈F for any x ∈X.
2. (Reproducing property) For any f ∈F and any x ∈X, f (x) = 〈f,k (·,x)〉F.
Theorem 1. Let F be a Hilbert function space over X. The following are equivalent:
1. F has a reproducing kernel.
2. For any x ∈X, the function δx : F 7→ R defined by δx (f) = f (x) is continuous.
Definition 2. Let F be a Hilbert function space over X. We say F is a reproducing kernel Hilbert space
over X if it satisfies the conditions of the previous theorem.
Proof. ( (1) ⇒ (2) ) . Suppose that F has a reproducing kernel, k : X ×X 7→ R. We need to show that if
(fn)
∞
n=1 converges to f ∈F, i.e., ‖fn −f‖F → 0, then for every x ∈X , |δx (fn) −δx (f)| = |fn (x) −f (x)|→
0. Now
δx (f) = f (x) = 〈f,k (·,x)〉F = 〈 lim
n→∞
fn,k (·,x)〉F
(a)
= lim
n→∞
〈fn,k (·,x)〉F = lim
n→∞
fn (x) = lim
n→∞
δx (fn)
Note that the identity “(a)” is implied by the continuity of inner product operator of its first argument.
Before proving the reverse implication, we state a classical result from functional analysis named the Riesz
representation theorem.
Theorem 2 (Riesz representation theorem). Let F be a Hilbert space and L : F 7→ R be a linear functional
on F. Then L is continuous if and only if there exists Φ ∈F such that L (f) = 〈f, Φ〉F for any f ∈F.
1
2
( (2) ⇒ (1) ) . Let x ∈ X . It’s trivial to see that δx is a linear functional, so due to the assumed
continuity of δx, Theorem 2 ensures the existence of a Φx ∈ F such that δx (f) = f (x) = 〈f, Φx〉F. Now
define k (x2,x1) := Φx1 (x2) ∈ R. Note that k (·,x) = Φx ∈F (establishing the first property of reproducing
kernels). Also,
f (x) = 〈f, Φx〉F = 〈f,k (·,x)〉F
which shows that the reproducing property holds.
The next result establishes that kernels and reproducing kernels are the same.
Theorem 3. Let k : X ×X 7→ R. Then k is a kernel if and only if k is a reproducing kernel of some RKHS
F over X.
Proof. (⇐) . We first prove the reverse implication. Let k be the reproducing kernel of F and define Φ :
X 7→F by Φ (x) = k (·,x). Now for any x′ ∈X , consider the function fx′ = k (·,x′). Using the reproducing
property and symmetry of the inner product, we obtain
k (x,x′) = fx′ (x) = 〈fx′,k (·,x)〉F = 〈k (·,x′) ,k (·,x)〉F
= 〈Φ (x′) , Φ (x)〉F = 〈Φ (x) , Φ (x′ ...
1 IntroductionThese notes introduces a particular kind of HiMartineMccracken314
1 Introduction
These notes introduces a particular kind of Hilbert space known as a reproducing kernel Hilbert space (RKHS).
We will establish connections with kernels, defined previously, and show that kernels and RKHSs are in one-to-
one correspondence. This material is largely drawn from Chapter 4 of [1], although some results are presented
in a slightly different way to ease digestion.
2 Reproducing Kernel Hilbert Spaces
Throughout these notes, we use the term “Hilbert function space over X ” to refer to a Hilbert space whose
elements are functions f : X 7→ R.
Definition 1. (Reproducing Kernel) Let F be a Hilbert function space over X . A reproducing kernel of F is a
function k : X × X 7→ R which satisfies the following two properties:
1. k (·,x) ∈F for any x ∈X.
2. (Reproducing property) For any f ∈F and any x ∈X, f (x) = 〈f,k (·,x)〉F.
Theorem 1. Let F be a Hilbert function space over X. The following are equivalent:
1. F has a reproducing kernel.
2. For any x ∈X, the function δx : F 7→ R defined by δx (f) = f (x) is continuous.
Definition 2. Let F be a Hilbert function space over X. We say F is a reproducing kernel Hilbert space
over X if it satisfies the conditions of the previous theorem.
Proof. ( (1) ⇒ (2) ) . Suppose that F has a reproducing kernel, k : X ×X 7→ R. We need to show that if
(fn)
∞
n=1 converges to f ∈F, i.e., ‖fn −f‖F → 0, then for every x ∈X , |δx (fn) −δx (f)| = |fn (x) −f (x)|→
0. Now
δx (f) = f (x) = 〈f,k (·,x)〉F = 〈 lim
n→∞
fn,k (·,x)〉F
(a)
= lim
n→∞
〈fn,k (·,x)〉F = lim
n→∞
fn (x) = lim
n→∞
δx (fn)
Note that the identity “(a)” is implied by the continuity of inner product operator of its first argument.
Before proving the reverse implication, we state a classical result from functional analysis named the Riesz
representation theorem.
Theorem 2 (Riesz representation theorem). Let F be a Hilbert space and L : F 7→ R be a linear functional
on F. Then L is continuous if and only if there exists Φ ∈F such that L (f) = 〈f, Φ〉F for any f ∈F.
1
2
( (2) ⇒ (1) ) . Let x ∈ X . It’s trivial to see that δx is a linear functional, so due to the assumed
continuity of δx, Theorem 2 ensures the existence of a Φx ∈ F such that δx (f) = f (x) = 〈f, Φx〉F. Now
define k (x2,x1) := Φx1 (x2) ∈ R. Note that k (·,x) = Φx ∈F (establishing the first property of reproducing
kernels). Also,
f (x) = 〈f, Φx〉F = 〈f,k (·,x)〉F
which shows that the reproducing property holds.
The next result establishes that kernels and reproducing kernels are the same.
Theorem 3. Let k : X ×X 7→ R. Then k is a kernel if and only if k is a reproducing kernel of some RKHS
F over X.
Proof. (⇐) . We first prove the reverse implication. Let k be the reproducing kernel of F and define Φ :
X 7→F by Φ (x) = k (·,x). Now for any x′ ∈X , consider the function fx′ = k (·,x′). Using the reproducing
property and symmetry of the inner product, we obtain
k (x,x′) = fx′ (x) = 〈fx′,k (·,x)〉F = 〈k (·,x′) ,k (·,x)〉F
= 〈Φ (x′) , Φ (x)〉F = 〈Φ (x) , Φ (x′ ...
The document discusses properties and differentiation of natural exponential functions:
1) The natural exponential function f(x) = ex is its own derivative. Taking the derivative of both sides of ln(ex) = x gives f'(x) = ex.
2) Geometrically, the slope of the graph of f(x) = ex at any point (x, ex) is equal to the y-value ex.
3) Examples are provided to demonstrate differentiating exponential functions, locating relative extrema of exponential functions, and showing that the standard normal probability density function has points of inflection at x = ±1.
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), Fourier series expansion of periodic functions, and Fourier transform properties such as linearity, translation and modulation. It also defines the Dirac delta function and discusses convolution theory and the Parseval relation.
This document discusses Fourier series, which represent periodic functions as an infinite series of sines and cosines. It defines periodic functions and their periods. Fourier series can be used to solve differential equations and represent many discontinuous periodic functions. The coefficients in a Fourier series are calculated using Euler-Fourier formulas. A function must meet certain conditions to have a Fourier series representation. Knowing if a function is even or odd can simplify determining the coefficients. Half-range Fourier series are also discussed as representing functions defined over half their period. Examples are provided to illustrate Fourier series concepts.
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
Basic Knowledge Representation in First Order Logic.pptAshfaqAhmed693399
This document provides an overview of basic knowledge representation in first-order logic (FOL). It discusses objects, properties, classes, and relations that can be modeled in FOL. It also covers the syntax of FOL, including predicates, terms, quantifiers, and scopes. Translation of English sentences to FOL formulas is demonstrated. Semantics such as domains, interpretations, models, validity, and logical consequence are defined. Representing change over time using the situation calculus is briefly discussed.
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.
The document provides an overview of the concept of derivatives. It states that a function is differentiable at a point if the slope of its tangent line at that point is well-defined. It also notes that a function is differentiable over an interval if it is differentiable at every point in the interval. The document then discusses how derivatives can be systematically calculated by taking the derivatives of basic functions like power, trigonometric, logarithmic and exponential functions, and understanding how derivatives behave under operations like addition, subtraction, multiplication, division and function composition.
1) Function notation y = f(x) denotes a functional relationship between variables x and y.
2) If a rule relates y to x, like y = 5x + 2, it can be written as the function f(x) = 5x + 2, where f(x) represents the value of the function for input x.
3) The domain is the set of x values, and the range is the set of f(x) values, with f(x) evaluating the function by substituting a value for x.
This document provides an overview of basic knowledge representation in first-order logic (FOL). It describes how FOL can be used to model objects, properties, classes, and relations in the world. It explains the syntax of FOL, including predicates, terms, quantifiers, and scopes. It also discusses translating English sentences to FOL representations and the semantics and model theory of FOL. Finally, it briefly introduces higher-order logic and the situation calculus for representing change over time.
The document discusses Fourier series and related concepts:
- Fourier series decompose a function into a weighted sum of sinusoids, with the weights determined by the Fourier transform.
- They can represent both real and complex functions as a sum of coefficients and a particular series.
- Fourier series generally converge everywhere except at discontinuities and converge to the function at almost every point, according to Dirichlet's condition.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses 11 properties of the Fourier transform: (1) Linearity and superposition, (2) Time scaling, (3) Time shifting, (4) Duality or symmetry, (5) Area under the time domain function equals the Fourier transform at f=0, (6) Area under the Fourier transform equals the time domain function at t=0, (7) Frequency shifting, (8) Differentiation in the time domain, (9) Integration in the time domain, (10) Multiplication in the time domain becomes convolution in the frequency domain, and (11) Convolution in the time domain becomes multiplication in the frequency domain. Each property is explained briefly.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
Some Characterizations of Riesz Valued Sequence Spaces generated by an Order ...UniversitasGadjahMada
In this paper, we introduce an order φ-function on a Riesz space. Further, we construct a Riesz valued sequence spaces by using the φ-function and obtain the condition to characterize these spaces.
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
In this PPT, you will learn about Gradient, Divergence and Curl of Function with examples.
Visit our YouTube Channel SR Physics Academy and watch the video on Gradient, Divergence and Curl of a Function.
The link to the YouTube Channel is given below:
https://www.youtube.com/channel/UC6TP5JDKFT6qAkOT1FGHuYQ
Step 1: Search for ‘SR Physics Academy’ on YouTube.
Step 2: Go to Playlist ’Electromagnetic theory in English’.
Step 3: Watch the video ‘Gradient, Divergence and Curl of a Function’.
laplace transform of function of the 풕^풏f(t)ABHIJITPATRA23
The document discusses the Laplace transform and some of its properties. It provides the definition of the Laplace transform and examples of finding the Laplace transform of elementary functions like t7 and e-5t. It also demonstrates using properties like shifting and multiplication to find the Laplace transform of more complex functions like te-t sin 3t. The conclusion states that Laplace transforms are a powerful tool used in many areas like mathematics, physics, and engineering.
This document summarizes key points from a lecture on Lévy processes and stochastic calculus. It discusses filtrations and how they allow consideration of events up to a given time. It defines Markov processes and their transition probabilities. It also discusses martingales and how Lévy processes with zero mean are martingales. Important martingale examples related to Lévy processes are presented.
This document announces the winners of the 2024 Youth Poster Contest organized by MATFORCE. It lists the grand prize and age category winners for grades K-6, 7-12, and individual age groups from 5 years old to 18 years old.
KALYAN MATKA | MATKA RESULT | KALYAN MATKA TIPS | SATTA MATKA | MATKA.COM | MATKA PANA JODI TODAY | BATTA SATKA | MATKA PATTI JODI NUMBER | MATKA RESULTS | MATKA CHART | MATKA JODI | SATTA COM | FULL RATE GAME | MATKA GAME | MATKA WAPKA | ALL MATKA RESULT LIVE ONLINE | MATKA RESULT | KALYAN MATKA RESULT | DPBOSS MATKA 143 | MAIN MATKA
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), Fourier series expansion of periodic functions, and Fourier transform properties such as linearity, translation and modulation. It also defines the Dirac delta function and discusses convolution theory and the Parseval relation.
This document discusses Fourier series, which represent periodic functions as an infinite series of sines and cosines. It defines periodic functions and their periods. Fourier series can be used to solve differential equations and represent many discontinuous periodic functions. The coefficients in a Fourier series are calculated using Euler-Fourier formulas. A function must meet certain conditions to have a Fourier series representation. Knowing if a function is even or odd can simplify determining the coefficients. Half-range Fourier series are also discussed as representing functions defined over half their period. Examples are provided to illustrate Fourier series concepts.
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
Basic Knowledge Representation in First Order Logic.pptAshfaqAhmed693399
This document provides an overview of basic knowledge representation in first-order logic (FOL). It discusses objects, properties, classes, and relations that can be modeled in FOL. It also covers the syntax of FOL, including predicates, terms, quantifiers, and scopes. Translation of English sentences to FOL formulas is demonstrated. Semantics such as domains, interpretations, models, validity, and logical consequence are defined. Representing change over time using the situation calculus is briefly discussed.
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.
The document provides an overview of the concept of derivatives. It states that a function is differentiable at a point if the slope of its tangent line at that point is well-defined. It also notes that a function is differentiable over an interval if it is differentiable at every point in the interval. The document then discusses how derivatives can be systematically calculated by taking the derivatives of basic functions like power, trigonometric, logarithmic and exponential functions, and understanding how derivatives behave under operations like addition, subtraction, multiplication, division and function composition.
1) Function notation y = f(x) denotes a functional relationship between variables x and y.
2) If a rule relates y to x, like y = 5x + 2, it can be written as the function f(x) = 5x + 2, where f(x) represents the value of the function for input x.
3) The domain is the set of x values, and the range is the set of f(x) values, with f(x) evaluating the function by substituting a value for x.
This document provides an overview of basic knowledge representation in first-order logic (FOL). It describes how FOL can be used to model objects, properties, classes, and relations in the world. It explains the syntax of FOL, including predicates, terms, quantifiers, and scopes. It also discusses translating English sentences to FOL representations and the semantics and model theory of FOL. Finally, it briefly introduces higher-order logic and the situation calculus for representing change over time.
The document discusses Fourier series and related concepts:
- Fourier series decompose a function into a weighted sum of sinusoids, with the weights determined by the Fourier transform.
- They can represent both real and complex functions as a sum of coefficients and a particular series.
- Fourier series generally converge everywhere except at discontinuities and converge to the function at almost every point, according to Dirichlet's condition.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses 11 properties of the Fourier transform: (1) Linearity and superposition, (2) Time scaling, (3) Time shifting, (4) Duality or symmetry, (5) Area under the time domain function equals the Fourier transform at f=0, (6) Area under the Fourier transform equals the time domain function at t=0, (7) Frequency shifting, (8) Differentiation in the time domain, (9) Integration in the time domain, (10) Multiplication in the time domain becomes convolution in the frequency domain, and (11) Convolution in the time domain becomes multiplication in the frequency domain. Each property is explained briefly.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
Some Characterizations of Riesz Valued Sequence Spaces generated by an Order ...UniversitasGadjahMada
In this paper, we introduce an order φ-function on a Riesz space. Further, we construct a Riesz valued sequence spaces by using the φ-function and obtain the condition to characterize these spaces.
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
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laplace transform of function of the 풕^풏f(t)ABHIJITPATRA23
The document discusses the Laplace transform and some of its properties. It provides the definition of the Laplace transform and examples of finding the Laplace transform of elementary functions like t7 and e-5t. It also demonstrates using properties like shifting and multiplication to find the Laplace transform of more complex functions like te-t sin 3t. The conclusion states that Laplace transforms are a powerful tool used in many areas like mathematics, physics, and engineering.
This document summarizes key points from a lecture on Lévy processes and stochastic calculus. It discusses filtrations and how they allow consideration of events up to a given time. It defines Markov processes and their transition probabilities. It also discusses martingales and how Lévy processes with zero mean are martingales. Important martingale examples related to Lévy processes are presented.
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2. Introduction
• “The profound study of nature is the most
fertile source of mathematical discoveries.”
---------------Joseph Fourier
3. Definition of the Fourier Transform
and Examples
• The Fourier transform of f(x) is denoted by F{f(x)} = F(k), k ∈ R, and defined
by the integral
where F is called the Fourier transform operator or the Fourier
transformation.
The inverse Fourier transform, denoted by 𝐹−1{F(k)} = f(x), is defined by
where 𝐹−1 is called the inverse Fourier transform operator.
10. Convolution Therorem
• The convolution of two integrable functions f(x) and g(x), denoted by (f ∗
g)(x), is defined by
• provided the integral in (2.5.10) exists, where the factor
1
2𝜋
is a matter of
choice.
13. The convolution has the following algebraic
properties:
• f ∗ g = g ∗ f (Commutative), (2.5.14)
• f ∗ (g ∗ h)=(f ∗ g) ∗ h (Associative), (2.5.15)
• (αf + βg) ∗ h = α (f ∗ h) + β (g ∗ h)
(Distributive), (2.5.16)
• f ∗ 2𝜋𝛿= f = 2𝜋𝛿∗ f (Identity), (2.5.17)
where α and β are constants.
25. Uses
Many linear boundary value and initial value problems in
applied mathematics, mathematical physics, and
engineering science can be effectively solved by the use of
the Fourier transform, the Fourier cosine transform, or the
Fourier sine transform. These transforms are very useful for
solving differential or integral equations.
the replacement of the continuous derivatives in the governing partial differential equations with equivalent finite difference expressions and the rearrangement of the resulting algebraic equation into an algorithm.
In practice the algebraic equations that result from the discretisation process, Sect. 3.1, are obtained on a finite grid. It is to be expected, from the truncation errors given in Sects. 3.2 and 3.3, that more accurate solutions could be obtained on a refined grid. These aspects are considered further in Sect. 4.4. However for a given required solution accuracy it may be more economical to solve a higher-order finite difference scheme on a coarse grid than a low-order scheme on a finer grid. This leads to the concept of computational efficiency which is examined in Sect. 4.5. An important question concerning computational solutions is what guarantee can be given that the computational solution will be close to the exact solution of the partial differential equation(s) and under what circumstances the computational solution will coincide with the exact solution. The second part of this question can be answered (superficially) by requiring that the approximate (computational) solution should converge to the exact solution as the grid spacings At, Ax shrink to zero (Sect. 4.1). However, convergence is very difficult to establish directly so that an indirect route, as indicated in Fig. 4.1, is usually followed. The indirect route requires that the system of algebraic equations formed by the discretisation process (Sect. 3.1) should be consistent (Sect. 4.2) with the governing partial differential equation(s). Consistency implies that the discretisation process can be reversed, through a Taylor series expansion, to recover the governing equation(s). In addition, the algorithm used to solve the algebraic equations to give the approximate solution, T, must be stable (Sect. 4.3). Then the pseudo-equation. is invoked to imply convergence. The conditions under which (4.1) can be made precise are given by the Lax equivalence theorem (Sect. 4.1.1). It is very difficult to obtain theoretical guidance for the behaviour of the solution on a grid of finite size. Most of the useful theoretical results are strictly only applicable in the limit that the grid size shrinks to zero. However the connections that are established between convergence (Sect. 4.1), consistency (Sect. 4.2) and stability (Sect. 4.3) are also qualitatively useful in assessing computational solutions on a finite grid.
For the equations that govern fluid flow, convergence is usually impossible to demonstrate theoretically. However, for problems that possess an exact solution, like the diffusion equation, it is possible to obtain numerical solutions on a successively refined grid and compute a solution error. Convergence implies that the solution error should reduce to zero as the grid spacing is shrunk to zero. For program DIFF (Fig. 3.13), solutions have been obtained on successively refined spatial grids, Ax =0.2, 0.1, 0.05 and 0.025. The corresponding rms errors are shown in Table 4.1 for s=0.50 and 0.30. It is clear that the rms error reduces like Ax2 approximately. Based on these results it would be a reasonable inference that refining the grid would produce a further reduction in the rms error and, in the limit of Ax (for fixed s) going to zero, the solution of the algebraic equations would converge to the exact solution. The establishment of numerical convergence is rather an expensive process since usually very fine grids are necessary. As s is kept constant in the above example the timestep is being reduced by a factor of four for each halving of Ax. In Table 4.1 the solution error is computed at t = 5000 s. This implies the finest grid solution at s = 0.30 requires 266 time steps before the solution error is computed. For the diffusion equation (3.1) with zero boundary values and initial value T (x, 0) = sin(nx), 0 ~ x ~ 1, the rms solution error lelrms is plotted against grid . spacing A x in Fig. 4.2. The increased rate of convergence (fourth-order convergence) for s =i, compared with other values of s~! (second-order convergence), is clearly seen, i.e. the convergence rate is like AX4 for s=i, and like AX2 otherwise. As will be demonstrated in Sect. 4.2, the superior convergence rate for s=i is to be expected from a consideration of the leading term in the truncation error. Typically, for sufficiently small grid spacings A x, A t, the solution error will reduce like the truncation error as deltax, deltat _>0.
This is the tendency for any spontaneous perturbations (such as round-off error) in the solution of the system of algebraic equations (Figs. 3.1 and 4.1) to decay. A stable solution produced by the FTCS scheme with s = 0.5 is shown in Fig. 3.15. A typical unstable result (s = 0.6) is shown in Fig. 4.3. These results have been obtained with Ll x = 0.1 and the same initial and boundary conditions as used to generate Fig. 3.15. It is clear from Fig. 4.3 that an unphysical oscillation originates on the line of symmetry and propagates to the boundaries. The amplitude of the oscillation grows with increasing time. The concept of stability is concerned with the growth, or decay, of errors introduced at any stage of the computation. In this context, the errors referred to are not those produced by incorrect logic but those which occur because the computer cannot give answers to an infinite number of decimal places. In practice, each calculation made on the computer is carried out to a finite number of significant figures which introduces a round-off error at every step of the computation. Hence the computational solution to (3.41) is not T't 1, but * r;+ 1, the numerical solution of the system of algebraic equations. A particular method is stable if the cumulative effect of all the round-off errors produced in the application of the algorithm is negligible. More specifically, consider the errors ej= Tj-*Tj (4.15) introduced at grid points (j, n), where j = 2, 3, ... , J -1 and n = 0, 1, 2. It is usually not possible to determine the exact value of the numerical error ej at the (j, n)-th grid point for an arbitrary distribution of errors at other grid points. However, it can be estimated using certain standard methods, some of which will be discussed in this section. In practice, the numerical solutions are typically more accurate than these estimates indicate, because stability analyses often assume the worst possible combination of individual errors. For instance, it may be assumed that all errors have a distribution of signs so that their total effect is additive, which is not always the case. It can be shown that, for linear algebraic equations produced by discretisation, the corresponding error terms satisfy the same homogeneous algebraic equations as the values of T. For instance, using the FTCS scheme (3.41) means that we are actually calculating *T~+ 1 using *T~ *T'! and *T'! so that J J-l' J J+l, (4.16) Substitution of (4.15) into (4.16), followed by application of (3.41), which applies since the exact solutions ofthe algebraic equations, Tj, satisfy the FTCS algorithm, yields the homogeneous algebraic equation (4.17) Assuming given boundary and initial values, the initial errors, eJ, j = 2, 3, ... , J -1, and the boundary errors e7 and ej, n = 0, 1, 2, ... for this equation, will all be zero. Unless some (round-off) error is introduced in calculating the value of Tj at some interior node, the resulting errors in the solution will remain zero. The two most common methods of stability analysis are the matrix method and the von Neumann method. Both methods are based on predicting whether there will be a growth in the error between the true solution of the numerical algorithm and the actually computed solution, i.e. including round-off contamination. An alternative interpretation of stability analysis is to suppose that the initial conditions are represented by a Fourier series. Each harmonic or mode in the Fourier series will grow or decay depending on the discretised equation, which typically furnishes a specific expression for the growth (or decay) factor for each mode. If a particular mode can grow without bound, the discretised equation has an unstable solution. This interpretation of stability (Richtmyer and Morton 1967, pp.9-13) is exploited directly in the von Neumann method of stability analysis (Sects. 4.3.4 and 4.3.5). The unbounded growth of a particular mode is still possible if the discretised equations are solved exactly, i.e. with no (round-off) errors being present. If (round-off) errors are introduced the same unstable nature of the discretised equations will cause unacceptable growth of the errors. Consequently the procedures for analysing the stability of the discretised equations are the same irrespective of the manifestation of the inherent stability or instability.
It is clear that as Llt tends to zero, Ej tends to zero and (4.13) coincides with the governing equation. Consequently (4.8) is consistent with the governing equation. In (4.14) all spatial derivatives have been converted to equivalent tIme derivatives. Using (4.12) it would be possible to express the truncation error in terms of the spatial grid size and derivatives only, as in (4.7). A comparison of (4.14) and (4.7) indicates that there is no choice of s that will reduce the truncation error of the fully implicit scheme to O(LlX4). It might appear from the above two examples that consistency can be taken for granted. However, attempts to construct algorithms that are both accurate and stable can sometimes generate potentially inconsistent discretisations, e.g. the DuFort-Frankel scheme, Sect. 7.1.2.
For explicit methods a single unknown, e.g. Tj + 1 , appears on the left hand side of the algebraic formula resulting from discretisation.
Effectively this scheme evaluates the spatial derivative at the average of the nth and (n + l}-th time levels, i.e. at the (n + 1/2)-th time level. If a Taylor expansion is made about (j, n + 1/2), (7.22) is found to be consistent with (7.1) with a truncation error of O(Llt2 , Llx2 ). This is a considerable improvement over the fully implicit and FTCS schemes that are only first-order accurate in time. A von Neumann stability analysis indicates that the Crank-Nicolson scheme is unconditionally stable, Table 7.1. A rearrangement of (7.22) gives the algorithm -0.5s Tj:!:t +(1 +s) Tj+1-0.5s Tj:t =0.5s Tj-1 +(I-s) Tj+0.5s Tj+1 , (7.23) which may be compared with (7.20). By considering all spatial nodes (7.23) produces a tridiagonal system of equations which can be solved efficiently using the Thomas algorithm Because ofthe second-order temporal accuracy, the Crank-Nicolson scheme is a very popular method for solving parabolic partial differential equations. The properties of the Crank-Nicolson scheme are summarised in Table 7.1. A generalisation of (7.22) can be obtained by writing (7.24) where A Tj + 1 = Tr 1 - Tj and 0 ~ 13 ~ 1. If 13 = 0 the FTCS scheme is obtained. If 13=0.5 the Crank-Nicolson scheme is obtained and if 13= 1.0 the fully implicit scheme is obtained. A von Neumann stability analysis of (7.24) indicates that a stable solution is possible for At ~ (X(I- 213) if 0 ~ 13 < 1/2 no restriction if 1/2~f3~ 1 . It may be noted that the Crank-Nicolson scheme is on the boundary of the unconditionally stable regime. For many steady flow problems it is efficient to solve an equivalent transient problem until the solution no longer changes (Sect. 6.4). However, often the solution in different parts of the computational domain approaches the steady-state solution at significantly different rates; the equations are then said to be stiff (Sect. 7.4). Unfortunately the Crank-Nicolson scheme often produces an oscillatory solution in this situation which, although stable, does not approach the steady state rapidly. Certain three-level (in time) schemes are more effective than the Crank-Nicolson scheme in this regard.
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