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.
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.
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.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
In unit 8L Sound and Hearing you are expected to:
build on your knowledge of sound and hearing
explain how sound travels through media (solid, liquid and gas)
give an explanation of how the human ear works,
find out about the harmful effects of loud noise and how loud noise can be reduced
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
Here are some Messages received from top Statisticians, Mathematicians and Computer Scientists across the Globe on the Occasion of our Cerebrations for World Statistics Day at Convergence'2010 on Oct 20-21; Ramajs College, Delhi University.
Here are some Messages received from top Statisticians, Mathematicians and Computer Scientists across the Globe on the Occasion of our Cerebrations for World Statistics Day at Convergence'2010 on Oct 20-21; Ramajs College, Delhi University.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
Ft3 new
1. An introduction to
Fourier Analysis
University of Delhi
Professor Ajay Kumar
Department of Mathematics
Delhi-110007
2. There are several natural phenomena
that are described by periodic
functions. The position of a planet in
its orbit around the sun is a periodic
function of time; in Chemistry, the
arrangement of molecules in crystals
exhibits a periodic structure. The
theory of Fourier series deals with
periodic functions.
3. We begin with the concept of
Fourier SeriesFourier Series
Periodic function
A function f(t) is said to have a period
T or to be periodic with period T if
for all t, f(t+T)=f(t), where T is a
positive constant. The least value of
T>0 is called the principal period or
the fundamental period or simply the
period of f(t).
4. The function has periods 2π, 4π, 6π,
all equal
.
Let
If f(x) has the period
then has the period T.
…… since
5. Let a functionf be declared on the
interval [0,T). The periodic expansion
defined by the formulaof f is
Periodic expansionPeriodic expansion
6.
7. Theorem:
Let f be continuous on
converges uniformly to f for all
.
Suppose that the series
9. The numbers an and bn are called the Fourier
coefficients of f. When an and
bn are given by (1)and(2) the trigonometric
series is called the Fourier series of the
function f. Below is an example of an arbitrary
function (the green function) which we
approximate with Fourier series of various
lengths. As you can see, the ability to mimic
the behavior of the function increases with
increasing series length, and the nature of the
fit is that the "spikier" elements are fit
better by the higher order functions.
10.
11. In mathematics, the question of whether
the Fourier series of a function
converges to the given function is
researched by a field known as classic
harmonic analysis, a branch of
pure mathematics. For most engineering
uses of Fourier analysis, convergence is
generally simply assumed without
justification. However, convergence is not
necessarily a given in the general case,
and there are criteria which need to be
met in order for convergence to occur.
18. would wonder how to define a similar
notion for functions which are L-periodic.
Assume that f(x) is defined and
integrable on the interval [-L,L]. Set
Remark.
We defined the Fourier series for
functions which are 2π -periodic, one
19. The function F(x) is defined and
integrable on [-π , π ].Consider the Fourier
series of F(x)
Using the substitution,
we obtain the following definition:
20. Let f(x) be a function defined and integrable on
[-L,L]. The Fourier series of f(x) is
.
Definition.
for
where
25. In what sense does the series on the right
converge, and if it does converge,
in what sense is it equal to f(x)? These
questions depend on the nature of the
function f(x). Considering functions
f(x) defined on R which satisfy a reasonable
condition like
∫
∞
∞−
| f(x) | dx < ∞
∫
∞
∞−
| f(x) |
2
dx< ∞
or
26. Since such functions cannot be
periodic ,we cannot really hope to expand
such function in terms of sin kx and cos
kx ,
k = 0,1, 2, --.As in
the periodic case. Taking the clue
that the functions einx
correspond to the
homomorphism χn of S1 into S1 , we look
for continuous homomorphism R into
S1.It turns out that all such continuous
homomorphism are given by
27. For an integrable function x(t) , define
the Fourier transform by
every real number w.
The independent variable t represents
time, the transform variable ω represents
angular frequency .Other notations for
this same function are:
The function is complex-valued in general.
and
28. If is defined as above, and
is sufficiently smooth, then it can be
reconstructed by the inverse transform:
for every real number t
.
29. For a scalar random variable X the
characteristic function is defined as the
expected value of eitX, and t ∈ R is the
argument of the characteristic function:
Here FX
is the cumulative distribution function of X, If
random variable X has a probability density function ƒX
,
then the characteristic function is its Fourier transform.
30. In this section, all the results are derived
for the following definition (normalization)
of the Fourier transform:
Let's see how we compute a Fourier
Transform: consider a particular function
f(x) defined as
37. The second integrand is odd, so integration
over a symmetrical range gives 0.The value of
the first integral is given by Abramowitz and
Stegun ,so
a Gaussian transforms to another Gaussian
The Fourier transform of the Gaussian
function is another Gaussian:
38.
39. Note that the width sigma is oppositely
positioned in the arguments of the exponentials
This means the narrower a Gaussian is in one
domain, the broader it is in the other domain.
The Fourier transform can also be extended to
the space integrable functions defined on
where,
and is the space of
continuous functions on
.
40. Inthis case the definition usually appears as
and
is the inner product of the two vectors ω and x.
One may now use this to define the continuous
Fourier transform for compactly supported
smooth functions, which are dense in
where
41. The Plancherel theorem and
Parseval's theorem
It should be noted that depending on the
author either of these theorems might be
referred to as the Plancherel theorem or as
Parseval's theorem.If f(t) and g(t) are
square-integrable and F(ω) and G(ω) are
their Fourier transforms, then we have
Parseval's theorem:
.
42. where the bar denotes complex
conjugation. Therefore, the Fourier
transformation yields an isometric
automorphism of the Hilbert space
43. The Plancherel theorem, a special case of
Parseval's theorem, states that
This theorem is usually interpreted as
asserting the unitary property of the
Fourier transform.
44. For reasonable functions f and h , we
define the convolution f * h by
The convolution is explained by the following
graphs
Convolution of Functions
then
45.
46.
47. 1. ( a f + bg )^ = a f^ + b g^
where a,b ε C
2. If g (x) = f (x+ u),
then g^(y) = exp( 2πi yu) f^(y)
3.If h(x) = exp(2πi ux),
then h^(y) = f^(y-u)
Basic Facts about the Fourier transform
For reasonable functions
48. 4. (f ΄)^(y) = 2πi y f^ (y) ,
where f ΄ is the derivative of f
49. The Fourier transform has
become a powerful analytical tool
in diverse fields of science. In
some cases, the Fourier
transform can provide a means of
solving unwieldy equations that
describe dynamic responses to
electricity, heat or light.
50. In other cases, it can identify the
regular contributions to a
fluctuating signal, thereby helping
to make sense of observations in
astronomy, medicine and
chemistry. Perhaps because of its
usefulness, the Fourier transform
has been adapted for use on the
personal computer.
51. Consider the heat flow in an infinite rod
where the initial temperature is given . In
other words, we look for the solution to
the initial –value problem, sometimes
called a Cauchy problem PDE
2
, 0t xxu u x tα= − ∞ < < ∞ < < ∞
IC
( ,0) ( )u x x xφ= − ∞ < < ∞
Application of Fourier transform:
Solution of an Initial -Value Problem
52. There are three basic steps in solving
this problem.
−∞ ∞
STEP 1(Transforming the problem)
Since the space variable x ranges from
transform of the PDE and IC with
respect to this variable x . Doing
this, we get
, we take the Fourierto
53. 2
[ ] [ ]
[ ( ,0)] [ ( )]
t xxF u F u
F u x F x
α
φ
=
=
and using the properties of the Fourier
transform, we have
2 2( )
( )
dU t
U t
dt
α ξ= −
54. (0) ( )U ξ= Φ
is the Fourier transform ofΦ φ
( ) [ ( , )]U t F u x t=
where
55. is nothing more than a constant in this
differential equation, so the solution to
this problem is
ξ
2 2
( ) ( ) t
U t e α ξ
ξ −
= Φ
Step 2
(Solving the transformed problem)
Remember the new variable
56.
Step 3 (Finding the inverse transform)
To find the solution u(x,t) , we merely
compute
2 2
1 1
( , ) [ ( , )] [ ( ) ]t
u x t F U t F e α ξ
ξ ξ− − −
= = Φ
Now using one of the properties of
convolution, we get
57. 2 2
1 1
( , ) [ ( )] [ ]t
u x t F F e α ξ
ξ− − −
= Φ ∗
2 2
( / 4 )1
( )
2
x t
x e
t
α
φ
α π
−
= ∗
2 2
( ) / 41
( )
2
x t
e d
t
ξ α
φ ξ ξ
α π
∞
− −
−∞
= ∫
(using tables)
This is the solution to our problem.
62. Consider nth order linear nonhomogeneous
ordinary differential equations with constant
coefficients
1
1 1 0
( ) ( )
n n
n n
Ly x f x
L a D a D a D a−
−
=
= + + − − − + +
d
D
dx
=are constants,
, 0,1, ,ia i n= −−−
Taking Fourier transform
63. 1
1 1 0
ˆˆ[ ( ) ( ) ( ) ] ( ) ( )n n
n na ik a ik a ik a y k f k−
−+ + − − − + + =
where
0
( )
n
r
r
r
P z a z
=
= ∑
ˆ( ) ˆ ˆˆ ˆ( ) ( ) ( ) ( )( )
( )
f k
y k f k q k f g k
P ik
= = = ∗
where
1
ˆ( )
( )
q k
P ik
=
( ) ( ) ( )y x f q x d
∞
−∞
= −∫ ζ ζ ζ
ˆˆ( ) ( ) ( )P ik y k f k=
64. Find the solution of the ordinary
differential equation
2
2
2
( ),
d u
a u f x x
dx
− + = −∞ ∞< <
Applying Fourier transform
2 2
ˆ( )
ˆ( )
( ) ( ) ( )
f k
u k
k a
u x f g x d
∞
−∞
=
+
= −∫ ζ ζ ζ
65. where
1
2 2
1 1
( ) ( ) exp( )
2
g x f a x
k a a
−
= = −
+
so ( ) ( )
a x
u x f e d
∞
− −
−∞
= ∫
ζ
ζ ζ
66. Solve the following ordinary
differential equation
'' '
2 ( ) ( ) ( ) 0u t tu t u t+ + =
Applying Fourier transform
2 '
ˆ ˆ2 ( ( )) 0
d
w u i f u t u
dw
− + + =
Or
2
ˆ ˆ ˆ2 (( ) ( )) 0
d
w u i i wu w u
dw
− + − + =
2
ˆ
ˆ2
ˆ( ) w
du
wu
dw
u w ce−
= −
=
67. 2
ˆ
ˆ2
ˆ( ) w
du
wu
dw
u w ce−
= −
=
Inverse Fourier transformation gives
2
( )
4
( )
t
u t De
−
=
68. The method of Fourier transform can be
used to solve integral equations
( ) ( ) ( ) ( )f t g x t dt f x u x
∞
−∞
− + =∫ λ
Where g(x)and u(x) are given functions
and is a known parameter.
Applying Fourier transform
λ
ˆ ˆˆ ˆ( ) ( ) ( ) ( )f k g k f k u k+ =λ
69. ˆ( )ˆ( )
ˆ( )
ˆ( )
( )
ˆ( )
ikx
u k
f k
g k
u k
f x e dk
g k
∞
−∞
=
+
=
+∫
λ
λ
In particular, if so that
then the solution
becomes
1
( )g x
x
=
ˆ( ) sgng k i kπ= −
ˆ1 ( )
( )
2 ( sgn )
ikx
u k e
f x dk
i kπ π
∞
−∞
=
−∫ λ
70. If , and so that ,λ = 1
1
( )
2
x
g x
x
= ÷ ÷
1
ˆ( ) ,g k
ik
=
the solution reduces to
'ˆ1 ( ) 1
( ) ( ( )) ( )
2 (1 ) 2
ikx
x ikxu k e
f x dk f u x f e e dk
ikπ π
∞ ∞
−
−∞ −∞
= =
+∫ ∫
' '
( ) ( )exp( )x
u x e u x
∞
−
−∞
= ∗ = ζ ζ −∫
71. Consider the solution of the Laplace
equation in the half plane
0, y 0xx yyu u x+ = −∞ < < ∞, ≥
with the boundary conditions
u(x,0)=f(x), x−∞ < < ∞
( , ) 0u x y → ,x y→ ∞ → ∞as
Apply Fourier transform with respect to x,
to obtain 2
2
ˆˆ 0
d u
ku
dy
− =
72. ˆˆ ˆ( ,0) ( ), ( , ) 0u k f k u k y= → as y→∞
ˆˆ( , ) ( )exp( )u k y f k k y= −
( , ) ( ) ( )u x y f k g x d
∞
−∞
= −ζ ζ∫
where
1
2 2
1
( ) ( )
k y y
g x f e
x yπ
−−
= =
+
73. ( )
( ) sin
x
f x e
f x x
=
=
1
[ ] ( )
2
i x
F f f x e dxξ
π
∞
−
−∞
= ∫
x → ∞
The major drawback of the Fourier
transform is that all functions can not be
transformed; for example, even simple
functions like
cannot be transformed, since the integral
does not exist. Only functions that
damp to zero sufficiently fast as
have transforms.
74. As a rule of thumb;the more
concentrated f(t) is, the more spread
out is F(ω). In particular, if we
"squeeze" a function in t, it spreads out
in ω and vice-versa; and we cannot
arbitrarily concentrate both the
function
and its Fourier transform.
76. Note the both functions have circular
symmetry. The atom is a sharp
feature, whereas its transform is a
broad smooth function.This
illustrates
the reciprocal relationship between a
function and its Fourier transform.
78. The molecule consists of seven
atoms. Its transform shows some
detail, but the overall shape is still
that of the atomic transform. We
can consider the molecule as the
convolution of the point atom
structure and the atomic shape. Thus
its transform is the product of the
point atom transform and the
atomic transform
79. If we think of concentration in terms of
f living entirely on a set of finite measure,
then we have the following beautiful result
of Benedicks: Let f be a nonzero square
integrable function on R. Then the Lebesgue
measures of the sets { x, f(x) ≠ 0 } and
{y, f^(y) ≠ 0 } cannot both be finite.
Benedicks’s theorem
80. ( For those who are not familiar with the
jargon of measure theory, a (measurable)
subset A of R is of finite measure, if it
can be covered by a countable union of
intervals Ik such that ( length of Ik)
< ∞ .). This result is a significant
generalization of the fact , well known to
engineers, that a nonzero signal cannot be
both time limited and band limited.
∑k
81. The rate at which a function decay at
infinity can also be considered a measure
of concentration. The following elegant
result of Hardy’s states that both f and
f^ cannot be very rapidly decreasing:
Suppose f is a measurable function on R
such that f(x) ≤ A exp(-απx│ │ 2
) and
│f^(y) ≤ B exp(-βπy│
2
) for some positive
constants A,B, α,β then, if
Hardy’s theorem:
82. i αβ > 1,then f must necessarily
be a zero function a.e.
ii αβ < 1, then there are infinitely many
linearly independent functions
iii αβ = 1, then f(x) = c exp(-απx
2
)
for some constant c.
83. For f ε L1
( R ),
∫∫R R
implies f = 0 a.e.
Beurling’ Theorem
│f(x) f^(y) exp( 2π xy ) dx dy││ │ │ │
< ∞
84. Generalizations of these
results have been obtained
for a variety of locally
compact groups like
Heisenberg groups, Motion
groups, non-compact
connected semi-simple Lie
groups and connected
nilpotent Lie groups
85.
86. Jean Baptiste Joseph Fourier
(1768-1830)
Administrator, Egyptologist, engineer,
mathematician, physicist, revolutionary,
soldier, and teacher; incredible as it may
sound, Fourier was all these! Born on
March 21, 1768 in Auxerre, France,
orphaned at the age of ten, Fourier had a
brilliant school career in the local
Benedectine school in his home town.
87. His life was rich and varied: member and
president of the local revolutionary
committee during his youth, student at the
Ecole Normale, teacher at the Ecole
Polytechnique (where he succeeded the
great Lagrange in the chair of Analysis and
Mechanics), imprisoned several times thanks
to rapidly shifting political ideologies in
Paris, and finally as recognition of his
distinguished service to science, he was
elected permanent mathematical secretary
of the French Academy of Sciences in 1822
88. His mathematical and science achievements
are, of course, legion. He derived the partial
differential equation that governs heat
conduction, and, to study the problem of
heat conduction, systematically developed
the subject which later came to be known as
Fourier analysis- a subject which was in its
infancy during his time. Far transcending
the particular subject of heat conduction,
his work stimulated research in
mathematical physics, which has since been
often identified with the solution of
boundary-value problems,
89. encompassing many natural occurrences
such as sunspots, tides, and the weather.
His work also had a great influence on the
theory of functions of a real variable, one
of the main branches of modern
mathematics. Some of his great discoveries
are contained in his celebrated treatise
‘Théorie analytique de la chaleur’ published
in 1822. Fourier also made important
contributions to probability theory,
statistics, mechanics, optimization and
linear programming.
90. G.B.Folland and A.Sitaram
( J.Fourier Anal.Appl.(1997)
G.F.Price and A.Sitaram
( J.Functional Analysis (1988))
E.Kaniuth and A.Kumar
( Forum Mathematicum (1991))
E.Kaniuth and A.Kumar
( Math. Proc.Camb. Phil.Soc.(2001)
S.Thangavelu
(Colloquium Mathematicum (2001) and Math.Zeit.
(2001)
91. H.Dym and H.P.McKean, Fourier Series and
Integrals,Academic Press,1972.
R.Bhatia, Fourier Series, TRIM-2.Hindustan
Book Agency, 1993.
T.W. Korner,Fourier Analysis, Cambridge
University Press, U.K., 1988.
G.B.Folland, Fourier Analysis and its
Applications, Wadsworth and Brooks/ Cole,
U.S.A., 1992