I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found [here.](https://math.stackexchange.com/questions/1958253/cauchy-product-of-fourier-series-with-itself)

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be an $L^{\infty}$ function on $[0,2\pi)^d$, whose Fourier coefficients are real and are given by $\hat{f}$, and in particular $\hat{f}(0) = 0$. Suppose also that $\Gamma \subset \mathbb{R}^d$ is a rational lattice of full rank. I would like to compute the $L^4$-norm over $[0,2\pi)^d$ by using its Fourier series. Explicitly:

$$\displaystyle \int_{[0,2\pi)^d} |f(k)|^4 \mathrm{d}k = \int_{[0,2\pi)^d}\left( \sum_{a \in \Gamma} \hat{f}(a)e^{iak} \right)\left( \sum_{b \in \Gamma} \hat{f}(b)e^{-ibk} \right) \left( \sum_{c \in \Gamma} \hat{f}(c)e^{ick} \right)\left( \sum_{d \in \Gamma} \hat{f}(d)e^{-idk} \right)\mathrm{d}k,$$

where I have used the relation $|f|^4$ = $(|f|^2)^2$ = $f^2 \cdot \bar{f}^2$. Could anyone tell me what the resulting sum looks like, inside the integral? I'm not familiar with how to take Cauchy products of series whose indices range over a rational lattice. Essentially what I am trying to do is a repeat of the argument [here](https://math.stackexchange.com/questions/1957136/imposing-condition-on-a-cauchy-product), except with four copies of the Fourier series (up to conjugation) rather than two -- and then use the fact that

$$\displaystyle \int_{0}^{2\pi} e^{ink}\mathrm{d}k = \begin{cases} 
      0, & n\neq 0, \\
      2\pi, & n = 0, 
   \end{cases}$$

to impose a condition on the resulting sum. If $d = 1$, and we take the lattice $\Gamma$ to be $\mathbb{Z}$, and let the indices range from $0$ to $\infty$ then I've ended up with:

$$\displaystyle \int_{0}^{2\pi} \sum_{a = 0}^{\infty}\sum_{b=0}^{a}\sum_{c=0}^{b}\sum_{d=0}^{c} \hat{f}(d)\hat{f}(c-d)\hat{f}(b-c)\hat{f}(a-b)e^{i(2d - 2c + 2b - a)k}\mathrm{d}k,$$

from which we can impose the condition $2d - 2c + 2b - a = 0$ to get rid of the integral and introduce a factor of $2\pi$. But where can we go from here? I know that the Fourier coefficients for general dimension $d$ are:

$$\displaystyle \hat{f}(n) = (2\pi)^{d/2}\rho^{d/2}\|n\|^{-d/2}J_{d/2}(\rho \|n\|),$$

where $\| \cdot \|$ denotes the Euclidean norm, $\rho > 0$ is a constant, and $J_{\nu}$ is the Bessel function of the first kind. How can we simplify the sum from here?

For background information on where this problem comes from, see [this paper](http://www.homepages.ucl.ac.uk/%7Eucahlep/Papers/BSPFULL.pdf) (in particular, pages 10-11, 15-16). The paper considers $\sigma_{p}$ for $p = 1,2$, and this is my attempt to generalise that work to the case $p = 4$ and for general $p.$ The goal is to bound the $L^4$-norm.