On the positive definiteness of functions of the form $h(\rho(x))+\beta\rho(x)h'(\rho(x))$
V. P. Zastavnyi Donetsk State University
Abstract:
In the paper, the following problem is considered for positive definite functions on
$\mathbb{R}^n$ (the class
$\Phi(\mathbb{R}^n)$). Let a function
$h$ be continuous on
$[0,+\infty)$, differentiable on the interval
$(0,+\infty)$,
$th'(t)\to 0$ as
$t\to+0$,
$h(t)\not\equiv h(0)$, and
$h(\rho(x))\in\Phi(\mathbb{R}^n)$. Here the function
$\rho$ is continuous on
$\mathbb{R}^n$,
$\rho(x)>0$ for
$x\ne0$, and
$\rho(tx)=|t|\rho(x)$, for
$x\in \mathbb{R}^n$,
$t\in\mathbb{R}$. For
$\beta\in\mathbb{R}$, we define the function
$H_\beta(t):=h(t)+\beta th'(t)$ for
$t>0$ and set
$H_\beta(0):=h(0)$. It is required to find the set of
$\beta\in\mathbb{R}$ for which
$H_\beta(\rho(x))\in\Phi(\mathbb{R}^n)$. Under the above assumptions, this set is a closed interval $[-\beta(h,\mathbb{R}^n,\rho), \widetilde{\beta}(h,\mathbb{R}^n,\rho)]$ which contains the point
$0$. In Theorem 1, formulas for the ends of this closed interval are found. In the case of the Euclidean norm, when
$(\mathbb{R}^n,\rho)=\ell_{2}^{n}$, in Theorem 2, for a wide class of functions
$h$, the exact value for the right end is found:
$\widetilde{\beta}(h,\ell_{2}^{n})=1/n$. In Theorem 3, for the function
$h_p(t)=\exp(-t^p)$ in the case of
$(\mathbb{R}^n,\rho)=\ell_{q}^{n}$, exact values for the right end and, in several cases, for the left one are found: if
$0<p\leqslant q\leqslant 2$, then
$\widetilde{\beta}(h_p,\ell_{q}^{n})=1/n$, ${\beta}(h_q,\ell_{q}^{n})={\beta}(h_q,\ell_{q}^{1})/n$,
${\beta}(h_1,\ell_{1}^{n})=1/n$,
${\beta}(h_1,\ell_{2}^{n})=1$, and
${\beta}(h_2,\ell_{2}^{n})=0$.
Keywords:
positive definite function, completely monotone function, Schoenberg
problem.
UDC:
517.5+
519.213
MSC: 42A82 Received: 14.12.2024
Revised: 17.01.2025
DOI:
10.4213/mzm14593
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