Question about the Weiner-Khinchin theorem

[mrMeister]

Homework Statement:

The question is attached below.

Relevant Equations:

correlation equations
Wiener-Khinchin theorem

My try:

I tried to take the expression for the decomposition coefficients and put it into the equation that I had to prove.
Then, I tried to work with the integral limits in order to get into Wiener-Khinchin theorem or maybe Fourier transform of delta function but I didn't see any success in those tries.

It would be nice to get a new point of view.

Thank you.

EDIT: I did solve this problem eventually, but I did not use the relation for z(t) - how is that?

 

[marcusl]

Please shoe your work and also where you got stuck.

 

[mrMeister]

Watch my comment - I have solved this question but without the use of the relation for z(t).
I wanted to ask if it is hidden somewhere throughout the solution.

 

[marcusl]

So you want someone to tell you if you assumed anything in a derivation that you refuse to share? I don’t think you understand how PF works.