# Connecting First Passage Time to Power Spectrum

Let $f$ be a real function. Is there a connection between

• The first positive abscissa for which its autocorrelation function is equal to zero (which I call the first passage time, fpt)
• The largest frequency of $f$'s power spectrum

For the matter, we can assume $f$ infinitely differentiable, square-integrable and anything more if needed. I know the Wiener-Khinchin theorem relates autocorrelation to power spectrum, but I'm not sure how to go further.

• wait you ask $f$ a Schwartz function ? then ... no the first zero-crossing are not related to the spectrum, but a relevant example might be boring to find. Do you have a practical problem in mind ? – reuns Jul 12 '16 at 9:22
• Not really. In my case $f$ is a cubic spline, and I want to quantify its wigglyness. The fpt is one way to do that – yannick Jul 12 '16 at 11:09
• Never heard about the wigglyness. Can you elaborate ? (I know a lot on signal processing) – reuns Jul 12 '16 at 11:18
• By wigglyness I meant a characteristic lengthscale, what in physics we call correlation length or persistence length en.wikipedia.org/wiki/… – yannick Jul 12 '16 at 11:57
• You should explain your precise problem (with your cubic splines), because until now I don't understand it – reuns Jul 12 '16 at 12:11