Let {Cn} be the complex Fourier coefficients of f, 27-periodic, i.e. coefficients wrt the orthonormal set {einæ}. Express the Fourier coefficients of f (x-a), where a E R, in terms of Cn. (Note the similarity with the shift rule of the Laplace transform.)
Let {Cn} be the complex Fourier coefficients of f, 27-periodic, i.e. coefficients wrt the orthonormal set {einæ}. Express the Fourier coefficients of f (x-a), where a E R, in terms of Cn. (Note the similarity with the shift rule of the Laplace transform.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Let {Cn} be the complex Fourier coefficients of f, 27-periodic, i.e. coefficients wrt
the orthonormal set {ein}. Express the Fourier coefficients of f(x-a), where a E R,
in terms of cn. (Note the similarity with the shift rule of the Laplace transform.)
2. Let f, g : R → R be continuous and 27-periodic. For x E R, we define h(x) as
1
h(x)
25 / f(x – t)g(t)dt.
Show that h is 27-periodic, and compute the Fourier coefficients of h with respect
to those of f and g.
You may use: If F(y, z) is continuous and integrable on R? then
F(y, z)dydz = | | F(y, z)dzdy.
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