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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)


Solution Summary

A periodic train of Dirac functions are investigated via convolution. The solution is detailed and well presented.