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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![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 -→](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde1132a6-9684-48be-960f-3f9ee5658757%2Fea81334b-1519-4239-8911-94860174b236%2Fg5iq1q_processed.png&w=3840&q=75)
Transcribed Image Text: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 -→
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