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I have stumbled upon the following three-dimensional series: $$\Lambda_p = \sum_{\underline{n}} \left(\frac{\left|n_1\right|}{\left|\left|\underline{n}\right|\right|_2}\right)^p \left|\hat{f}(\underline{n})\right|^2$$ where the sum over $\underline{n}$ mean summing over $(n_1,n_2,n_3)\in \left(\mathbb{Z}^3\right)^*$ and $\hat{f}$ denotes three-dimensional Fourier transform of radial real-valued function $f$. $\Lambda_0$ can be computed by direct application of Parseval identity. $\Lambda_2$ is the third of $\Lambda_0$. What about other $p\in \mathbb{N}^*$? I am particulaly interested in the case $p=4$.

In my particular context, $f$ is the indicator function of a ball of radius $r$ centered on $0$, but the radiality property is probably is the most important thing to know about it.

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  • $\begingroup$ how do you select the vectors $\mathbf{n}$? $\endgroup$
    – Carlo Beenakker
    Commented May 18, 2022 at 8:11
  • $\begingroup$ Sorry for being lazy; I explicited the summation. $\endgroup$
    – CNS
    Commented May 18, 2022 at 11:04
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    $\begingroup$ What are you hoping for? Obviously it is decreasing in $p$ (I assume you mean $|n_1|$, or $p$ even) so you already have an upper bound with $p=2$. You expect more than an upper estimate, e.g. a continuous equivalent formula, or something else? $\endgroup$
    – username
    Commented May 18, 2022 at 11:53
  • $\begingroup$ I'm probably too optimistic but I expect $\Lambda_4$ to have a closed-form expression. $\endgroup$
    – CNS
    Commented May 18, 2022 at 12:06

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