Thread: Proofs about convolutions

The problem has 5 parts and is based on:


Let f,g L^2(T, dx). Define the convolution of the two functions by: (f ** g)(x) = 1/(2pi)


a) Prove that f**g is a continuous function.
b) Prove that f**g = g**f
c) Denote by h(n) = 1/(2pi) the Fourier coefficients of a function h L^2(T, dx). Prove that f(*^*g)(n) = , n Z
d) Deduce that, for three functions in L^2: f**(g**h) = (f**g)**h.
e) Is there a function e L^2(T, dx) with the property: e*f = f, f L^2(T,dx)?


Where ** is the convolution symbol and T = {z such that |z| = 1} = { | t [0, 2pi]}


Any help at all will be amazing. I have no idea what is going on.

For the first question use density of the continuous functions. For the second one, a substitution, using translation invariance.

For the second question, do u mean something like:
t --> (x-t)
(x-t) --> t x - (x-t) -t
?

Yes, that's what I mean.