Question about the Weiner-Khinchin theorem

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SUMMARY

The discussion centers on the Weiner-Khinchin theorem and the challenges faced in applying its concepts to a specific problem involving decomposition coefficients and integral limits. The original poster initially struggled with integrating the theorem into their solution but later resolved the issue without utilizing the relation for z(t). The conversation highlights the importance of sharing derivations and assumptions in mathematical discussions for clarity and collaborative problem-solving.

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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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Last edited:
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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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