Fourier Series/Transformations and Convolution

[Yosty22]

Homework Statement

(f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy
f(x) = ∑c_ne^inx
g(x) = ∑d_ne^inx

e_n is defined as the Fourier Coefficients for (f*g) {the convolution} an is denoted by:

e_n = 1/(2pi) integral from -pi to pi of (f*g)e^-inx dx

Evaluate e_n in terms of c_n and d_n

Hint: somewhere in the integral, the substitution z = x - y might be helpful.

Homework Equations

For convolution, if C(x) = the integral from -infinity to infinity of (f(y)g(x-y))dx, the Fourier transform of C(x) is equal to the Fourier transform of f times the Fourier transform of g.

The Attempt at a Solution

I'm really not too sure where to begin with this other than understanding the definition in the Relevant Equations section. Before this problem, all we were told was this definition. I do not see any way this is related to the Fourier series mentioned in the problem, let alone how to incorporate just the coefficients of the series into the integral.

Any help to point me in the right direction would be greatly appreciated.

 

[vela]

Do you understand how that result, the theorem you mention in the relevant equations section, was obtained? If not, start there. You might try something similar in solving the problem.

 

[Yosty22]

I am pretty confused. The only thing jumping out to me on what to do is to substitute substitute (f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy into the integral for e_n and work it like a double integral. I am not exactly too sure if this would work though because I cannot think of where c_n and d_n would come in though (since he wants the answer in terms of those coefficients).