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    • An introduction to Fourier Analysis University of Delhi Professor Ajay Kumar Department of Mathematics Delhi-110007
    • 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.
    • We begin with the concept of Fourier 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).
    • The function has periods 2π, 4π, 6π, all equal . Let If f(x) has the period then has the period T . …… since
    • Let a function f be declared on the interval [0, T ). The periodic expansion defined by the formula of f is Periodic expansion
    • Theorem: Let f be continuous on   converges uniformly to f for all . Suppose that the series
    •   Then (2) (1)
    • The numbers a n and b n are called the Fourier coefficients of f . When a n and b n 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.
    • 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.
    • Find the Fourier series of the function Hence Example.
    • Find the Fourier series of the function The Fourier series of f ( x ) is Example.
    • Find the Fourier series of the function Example.
    • 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
    • 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:
    • Let f ( x ) be a function defined and integrable on [- L , L ]. The Fourier series of f ( x ) is . Definition. for where
    • Find the Fourier series of Example.
    • the Fourier series of f ( x ) can be written in complex form as   where   Using Euler's identities,
    • and n =1,2,… n =1,2,…
    • 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
    • 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 e inx correspond to the homomorphism χ n of S 1 into S 1 , we look for continuous homomorphism R into S 1 .It turns out that all such continuous homomorphism are given by φ λ (x) = exp (iλx).
    • 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  
    • If is defined as above, and is sufficiently smooth, then it can be reconstructed by the inverse transform : for every real number t .
    • For a scalar random variable X the characteristic function is defined as the expected value of e itX , and t ∈ R is the argument of the characteristic function: Here F X is the cumulative distribution function of X , If random variable X has a probability density function ƒ X , then the characteristic function is its Fourier transform .
    • 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
    • Its Fourier transform is:
    • In this case F ( u ) is purely real, which is a consequence of the original data being symmetric in x and - x . A graph of F ( u ) is shown in Fig.
    • Fourier Transform--Gaussian The Fourier transform of a Gaussian function is given by
    • 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:
    • 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 .
    • In this 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
    • 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 : .
    • where the bar denotes complex conjugation . Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space
    • 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.
    • 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
    • 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
    • 4. (f ΄)^(y) = 2πi y f^ (y) , where f ΄ is the derivative of f
    • 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.
    • 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.
    • 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 IC Application of Fourier transform: Solution of an Initial -Value Problem
    • 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 Fourier to
    • and using the properties of the Fourier transform, we have
    • is the Fourier transform of where
    • is nothing more than a constant in this differential equation, so the solution to this problem is
        • Step 2
      • (Solving the transformed problem)
      Remember the new variable
      • Step 3 (Finding the inverse transform)
      • To find the solution u(x,t) , we merely compute
      Now using one of the properties of convolution, we get
    • (using tables) This is the solution to our problem.
    • as Let in -periodic.
    • So,
    • Consider nth order linear nonhomogeneous ordinary differential equations with constant coefficients are constants, Taking Fourier transform
    • where where
    • Find the solution of the ordinary differential equation Applying Fourier transform
    • where so
    • Solve the following ordinary differential equation Applying Fourier transform Or
    • Inverse Fourier transformation gives
    • The method of Fourier transform can be used to solve integral equations Where g(x)and u(x) are given functions and is a known parameter. Applying Fourier transform
    • In particular, if so that then the solution becomes
    • If , and so that , the solution reduces to
    • Consider the solution of the Laplace equation in the half plane with the boundary conditions as Apply Fourier transform with respect to x, to obtain
    • as where
    • 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.
    • As a rule of thumb;the more concentrated f ( t ) is, the more spread out is F (ω). In particular, if we &quot;squeeze&quot; a function in t , it spreads out in ω and vice-versa; and we cannot arbitrarily concentrate both the function and its Fourier transform.
    • An atom, and its Fourier Transform:
    • 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.
    • A molecule, and its Fourier Transform:
    • 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
    • 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
    • ( 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 I k such that ( length of I k ) < ∞ .). 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.
    • 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 :
    • 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.
    • For f ε L 1 ( R ), implies f = 0 a.e. Beurling’ Theorem │ f(x) ││f^(y) │exp( 2π │xy│) dx dy < ∞
    • 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
    • Joseph Fourier
    • 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.
    • 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
    • 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,
    • 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 ‘ Th é orie analytique de la chaleur’ published in 1822. Fourier also made important contributions to probability theory, statistics, mechanics, optimization and linear programming.
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      • G.B.Folland, Fourier Analysis and its Applications, Wadsworth and Brooks/ Cole, U.S.A., 1992
    • Delhi Chennai Bangalore Mumbai Kolkata Vallabhbhai Vidya Nagar Kanpur Harmonic Analysis in India
    • Thank You