Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite function.

Define the set $S$ = $\{x_1,x_2,...x_n\}$ containing $n$ distinct points in $\mathbb{R}^N$

Define matrices $F$ and $G$ as $F = [f_{i,j}]_{n\times n}$ and $G = [g_{i,j}]_{n\times n}$, where $f_{i,j} = f(x_i-x_j)$ and $g_{i,j} = g(x_i-x_j)$.

Question : What is the relation between the matrices $F$ and $G$? If there isn't any simple relation, then how can I determine matrix $G$ from matrix $F$?

  • $\begingroup$ Is this question already solved? If someone has any quick reference works or a book, appreciate it. $\endgroup$ – Rajesh Dachiraju Jan 29 at 6:21
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    $\begingroup$ I doubt there is any link between $F$ and $G$. Just think about $n = 1$: the quantities $f(0)$ and $g(0) = f * f(0)$ are not related in any meaningful way. $\endgroup$ – Mateusz Kwaśnicki Jan 29 at 13:26

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