Problem 2.19 This problem is designed to guide you through a “proof" of Plancherel's theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to infinity. (a) Dirichlet's theorem says that "any" function f (x) on the interval [-a, +a]can be expanded as a Fourier series: (). NA X NT X S (x) = E[a, sin (*) + b, cos ( %3D n=0 Show that this can be written equivalently as f (x) = E Cneinax/a_ n=-00 What is Cn, in terms of an and b„? (b) Show (by appropriate modification of Fourier's trick) that 1 •+a | f(x)e¯inxx/adx. Cn 2a (c) Eliminate n and c„ in favor of the new variables k = (nx /a) and F (k) = /2/Tac,. Show that (a) and (b) now become +a 1 f (x) = E F(k)eik* Ak; 1 F (k) = 2л 2T $ (x)e¯iks dx, n=-00 where Ak is the increment in k from one n to the next. (d) Take the limit a → ∞ to obtain Planchereľ's theorem. Comment: In view of their quite different origins, it is surprising (and delightful) that the two formulas-one for F(k) in terms of f (x), the other for f (x) in terms of F(k)-have such a similar structure in the limit a -→

Problem 2.19 This problem is designed to guide you through a “proof" of Plancherel's theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to infinity. (a) Dirichlet's theorem says that "any" function f (x) on the interval [-a, +a]can be expanded as a Fourier series: (). NA X NT X S (x) = E[a, sin () + b, cos ( %3D n=0 Show that this can be written equivalently as f (x) = E Cneinax/a_ n=-00 What is Cn, in terms of an and b„? (b) Show (by appropriate modification of Fourier's trick) that 1 •+a | f(x)e¯inxx/adx. Cn 2a (c) Eliminate n and c„ in favor of the new variables k = (nx /a) and F (k) = /2/Tac,. Show that (a) and (b) now become +a 1 f (x) = E F(k)eik Ak; 1 F (k) = 2л 2T $ (x)e¯iks dx, n=-00 where Ak is the increment in k from one n to the next. (d) Take the limit a → ∞ to obtain Planchereľ's theorem. Comment: In view of their quite different origins, it is surprising (and delightful) that the two formulas-one for F(k) in terms of f (x), the other for f (x) in terms of F(k)-have such a similar structure in the limit a -→

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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