On an extremal problem for positive definite functions

A. D. Manov

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 161-171

Résumé

In this paper we consider an extremal problem related to a set of continuous positive definite functions on $\mathbb{R}$ whose support is contained in the closed interval $[-\sigma,\sigma]$, $\sigma>0$ and the value at the origin is fixed (the class $\mathfrak{F}_\sigma$). We consider the following problem. Let $\mu$ be a linear locally bounded functional on the set of continuous functions which have compact support, i.e. $C_c(\mathbb{R})$ and suppose that $\mu$ is real-valued functional on the sets $\mathfrak{F}_\sigma$, $\sigma>0$. For a fixed $\sigma>0$, it is required to find the following constants: $$ M(\mu,\sigma):=\sup\left\{ \mu(\varphi): \varphi\in\mathfrak{F}_\sigma\right\},\ m(\mu,\sigma):=\inf\left\{ \mu(\varphi): \varphi\in\mathfrak{F}_\sigma\right\}. $$ We have obtained a general solution to this problem for functionals of the following form $\mu(\varphi)=\int_\mathbb{R}\varphi(x)\rho(x)dx$, $\varphi\in C_c(\mathbb{R})$, where $\rho\in L_{loc}(\mathbb{R})$ and $\rho(x)=\overline{\rho(-x)}$ a.e. on $x\in\mathbb{R}$. For $\rho(x)\equiv1$, the value of $M(\mu,\sigma)$ was obtained by Siegel in 1935 and, independently, by Boas and Kac in 1945. In this article, we have obtained explicit solution to the problem under consideration in cases of $\rho(x)=ix$, $\rho(x)=x^2$ and $\rho(x)=i\mathop{\rm sign} x$, $x\in\mathbb{R}$. In addition, in this paper we study the connection between the problem under consideration and pointwise inequalities for entire functions of exponential type $\leqslant\sigma$ whose restrictions on $\mathbb{R}$ are in $L_1(\mathbb{R})$. In particular, sharp inequalities are obtained for the first and second derivatives of such functions.

Détail

Keywords: positive-definite functions, extremal problems, Bochner theorem, Fourier transform, entire functions of exponential type.