On some extremal problems for entire functions of exponential type
A. D. Manovab a Steklov Mathematical Institute of Russian Academy of Sciences (Moscow)
b Donetsk State University (Donetsk)
Abstract:
In this paper we consider a number of extremal problems for nonnegative and integrable entire functions of exponential type
$\leqslant\sigma$ (the class
$\mathcal{E}_{1,\sigma}^+$).
The problems under consideration have the following form. Let
$\Lambda_\rho$ be a translation invariant operator with a locally integrable symbol
$\rho(x)$,
$x\in\mathbb{R}$, such that
$\rho(x)=\overline{\rho(-x)}$,
$x\in\mathbb{R}$. For a fixed
$\sigma>0$, it is required to find the following constants:
\begin{equation*} \begin{split} M^{\ast}(\rho,\sigma)&=\sup\{(\Lambda_\rho f)(0):f\in\mathcal{E}_{1,\sigma}^{+},\ \|f\|_1=2\pi\},\\ m^{\ast}(\rho,\sigma)&=\inf\{(\Lambda_\rho f)(0):f\in\mathcal{E}_{1,\sigma}^{+},\ \|f\|_1=2\pi\}. \end{split} \end{equation*}
This general problem reduces to an equivalent extremal problem for positive-definite functions, the solution of which is known. As consequence, we obtained exact values of
$M^{\ast}(\rho,\sigma)$ and
$m^{\ast}(\rho,\sigma)$ for a number of different symbols
$\rho$. In particular, we consider cases where
$\Lambda_\rho$ is a differential or difference operator of a special form.
Keywords:
entire functions of exponential type, extremal problems, positive-definite functions, Bochner theorem, Fourier transform.
UDC:
517.5+
519.213 Received: 19.11.2024
Accepted: 10.03.2025
DOI:
10.22405/2226-8383-2025-26-1-47-61
Fulltext:
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