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# Fourier Analysis and Series : Periodic Functions and Convolution

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COMMENT FROM STUDENT:
My book says that you should be able to use f(x) = 1/2(x-t) in problem 2. Then f ~ 1/2 sum from negative infinity to positive ...

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Periodic functions and convolution are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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## Periodic Functions via Convolution

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"Periodic Function via Convolution"
Consider the periodic train of Dirac delta "functions"
f(x) =....
with real period ....
(a) FIND and DESCRIBE its Fourier transform F(k). What happens to F if c gets doubled?
(b) Let p(x + c) = p(x) be a periodic function.
Prove or disprove: p(x) is the convolution (i.e. f * g (x) E f(x ? t)g(t) dt) of a periodic train of Dirac delta functions with a non-periodic function, say g(x) in L2 (?co, oo). What is g(x)?
(c) Find the Fourier transform
5(k) FpJ(k) = * f ep(x) dx
of p(x) in terms of the Fourier transform (k) of that nonperiodic function. Relate your answer to the Fourier series coefficients of the periodic function p(x)

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