Abstract:
5pt Let $\Phi_n$ be the set of functions on $[0, +\infty)$ such that the corresponding radial function on $\mathbb R^n$ is positive definite. It is known that the functions in the class $\Phi_n\backslash\Phi_{n+1}$ have an infinite number of negative squares. With a function $f\in\Phi_n$ and a set of points $X_{n+2}:=\{x_1,\ldots,x_{n+2}\}\subset\mathbb{R}^{n+1}$, we associate the Schoenberg matrix $S_{X_{n+2}}(f)$. It is proved that, for a function $f\in\Phi_n$ and an $(n+1)$-dimensional regular simplex with the set of vertices $X_{n+2}$, the Schoenberg matrix $S_{X_{n+2}}(f)$ remains positive definite. Similar results are also obtained for a more general $(n+1)$-dimensional simplex. We also present configurations of $(n+3)$ points from $\mathbb{R}^{n+1}$ at which the corresponding Schoenberg matrix has one negative square.
Key words and phrases:Schoenberg matrix, positive definite function, radial positive definite function, simplex.
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