Relation between two matrices associated with a positive definite function

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 $$F$$?

• Is this question already solved? If someone has any quick reference works or a book, appreciate it. – Rajesh Dachiraju Jan 29 at 6:21
• 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. – Mateusz Kwaśnicki Jan 29 at 13:26