On an Extremal Problem for Compactly Supported Positive Definite Functions

Pub Date : 2024-05-13 DOI:10.1134/S1064562424701965

A. D. Manov

引用次数: 0

Abstract

An extremal problem for positive definite functions on \({{\mathbb{R}}^{n}}\) with a fixed support and a fixed value at the origin (the class \({{\mathfrak{F}}_{r}}({{\mathbb{R}}^{n}})\)) is considered. It is required to find the least upper bound for a special form functional over \({{\mathfrak{F}}_{r}}({{\mathbb{R}}^{n}})\). This problem is a generalization of the Turán problem for functions with support in a ball. A general solution to this problem for \(n \ne 2\) is obtained. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type.

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论紧凑支持正定函数的极值问题

摘要本文考虑的是\({\mathfrak{F}}_{r}}({\mathbb{R}}^{n}}\)上具有固定支撑和原点固定值的正定函数（类 \({\mathfrak{F}}_{r}}({\mathbb{R}}^{n}})）的极值问题。我们需要找到 \({{\mathfrak{F}}_{r}}({{\mathbb{R}}^{n}}) 上特殊形式函数的最小上界。）这个问题是对在球中有支持的函数的图兰问题的一般化。我们得到了这个问题对于 \(n \ne 2\) 的一般解。因此，得到了指数球型全函数导数的新的尖锐不等式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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