Strictly positive definite kernels on the circle
type: Item
instance of: scholarly article
title: Strictly positive definite kernels on the circle
zbMATH Open document ID: 0913.43002
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-05-31T00:00:00Z
review text: The author studies strictly positive definite and strictly conditionally negative definite functions on the circle \(S^1\). Strict positive definiteness, resp. strict conditional negative definiteness, means that strict inequalities hold whenever nontrivial bilinear forms of the corresponding matrices are considered. In \textit{Y. Xu} and \textit{E. W. Cheney} [Proc. Am. Math. Soc. 116, 977-982 (1992; Zbl 0787.43005)] it was shown that in the Schoenberg series representation \(f(t)= \sum^\infty_{k=0} =a_k \cos kt\) of the positive definite \(f\) one has strict positive definiteness provided all \(a_k>0\). Strict positive (and also strict conditional negative) definiteness will depend on the set of integers \(k\) for which \(a_k<0\) and not on the magnitude of the \(a_k\). In the present paper at first the result of Xu and Cheney is improved: Let \(f(t)= \sum_{k\in K} a_k\cos kt\) with \(a_k>0\) for all \(k\in K\) and \(\sum_{k\in K} a_k<\infty\). If \(K\) is a subset of \(\cup^\infty_{k=0} (b_k+ c_kN_k)\), where \(N_k= \{0,1, \dots, k\}\), \(b_k,c_k \in \mathbb{N}\) and \((c_k)\) is a prime sequence, then \(f\) is strictly positive definite. An increasing sequence \((c_k)\) of nonnegative integers is said to be prime if it is not contained in any set of the form \(p_1\mathbb{N} \cup p_2\mathbb{N} \cup\cdots \cup p_n \mathbb{N}\), where \(p_1, p_2, \dots, p_n\) are prime numbers. After that the author derives three independent necessary conditions for subsets \(K\subseteq \mathbb{N}_0\), where the functions \(f(t)\) with series representations over \(K\) as above are strictly positive definite. Each result for strict positive definiteness holds as well for strict conditional negative definiteness.
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 42A82
zbMATH DE Number: 878078
zbMATH Keywords: strictly positive definite and strictly conditionally negative definite functions
zbMATH Keywords: Schoenberg series representation
author: Valdir A. Menegatto
reviewed by: Rupert Lasser
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072211
OpenAlex ID: W1988771740
cites work: Q3755956
cites work: Interpolation by periodic radial basis functions
cites work: Strictly positive definite kernels on the hilbert sphere
cites work: Positive definite functions on spheres
cites work: Interpolation by periodic radial functions
cites work: Strictly Positive Definite Functions on Spheres
DOI: 10.1216/RMJM/1181072211
description: scientific article; zbMATH DE number 878078