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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 ...
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.
Periodic Functions via Convolution
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)