Question about the Weiner-Khinchin theorem

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The discussion revolves around attempts to apply the Wiener-Khinchin theorem and the Fourier transform to solve a problem involving decomposition coefficients. The original poster struggled with integrating limits and did not find success in their initial approach. They later resolved the problem but noted that they did not use the relation for z(t) in their solution. They seek clarification on whether the relation for z(t) is implicitly included in the solution and express confusion about the expectations for sharing derivations in the forum. Ultimately, the conversation highlights challenges in understanding the theorem's application and the sharing of problem-solving processes.
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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?
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Please shoe your work and also where you got stuck.
 
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.
 
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.
 

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