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.
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...