Please see the attached file for the fully formatted problems.
"Periodic Function via Convolution"
Consider the periodic train of Dirac delta "functions"
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)
A periodic train of Dirac functions are investigated via convolution. The solution is detailed and well presented.