Proofs about convolutions

The problem has 5 parts and is based on:

Let f,g $\displaystyle \in$ L^2(T, dx). Define the convolution of the two functions by: (f ** g)(x) = 1/(2pi) $\displaystyle \int_T f(x-t)g(t)dt$

a) Prove that f**g is a continuous function.

b) Prove that f**g = g**f

c) Denote by h(n) = 1/(2pi)$\displaystyle \int_Th(x)e^{-inx}dx$ the Fourier coefficients of a function h $\displaystyle \in$L^2(T, dx). Prove that f(*^*g)(n) = $\displaystyle \hat{f}(n)\hat{g}(n)$, n $\displaystyle \in$Z

d) Deduce that, for three functions in L^2: f**(g**h) = (f**g)**h.

e) Is there a function e $\displaystyle \in$ L^2(T, dx) with the property: e*f = f, f $\displaystyle \in$L^2(T,dx)?

Where ** is the convolution symbol and T = {z such that |z| = 1} = {$\displaystyle e^{it}$ | t $\displaystyle \in$[0, 2pi]}

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

Re: Proofs about convolutions

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

Re: Proofs about convolutions

For the second question, do u mean something like:

t --> (x-t)

(x-t) --> t x - (x-t) -t

?

Re: Proofs about convolutions