If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+...a_n \cos(2 \pi n \xi)$

Question

Let $f \in L^2 \cap C_c$ , then I want to show that

$$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$

Does anybody know this result or know how this can be shown? .

The problem is that $g$ is at first glance, only in $L^1[0,2\pi],$ so it is not even clear that this expansion exists (and not by no means it is clear to me that this representation is finite). Despite, the fact that only cos terms appear is clear, from the absolute value in the definition of $g$.

Answer 1

This is a consequence of the Poisson summation formula, specifically equation 3 of the article. The idea of Poisson summation is that the periodization of a function $F$, obtained by summing its translates, has a Fourier series whose coefficients are found by sampling $\hat F$. If $\hat F$ is compactly supported, this Fourier series becomes a trigonometric polynomial. In your situation, $F$ is $|\hat f|^2$. The Fourier transform of $|\hat f|^2$ is basically the convolution of $f$ with itself (modulo normalization details), so it is compactly supported.