Uniqueness of the Continuation of a Certain Function to a Positive Definite Function

作     者：Manov, A. D. 

作者机构：Donetsk Natl Univ UA-83114 Donetsk Ukraine 

出 版 物：《MATHEMATICAL NOTES》 (数学札记)

年 卷 期：2020年第107卷第3-4期

页      面：639-652页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：extension of positive definite functions Bochner-Khinchine theorem piecewise linear functions nonnegative trigonometric polynomials extremal problems 

摘      要：In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| a, can be extended to a positive definite one on R if and only if 0 a = 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| = a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a is an element of [1/2,1] and Re phi(x) = 1 - |x|, |x| = a, for some phi is an element of T, then phi(x) = (1 - |x|) (+), x G R. In addition, for any a G (0,1/2), there exists a function phi is an element of T such that phi(x) = 1 - |x|, |x| = a, but phi(x) not equal (1 - |x|)(+). Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.