On differentiability and analyticity of positive definite functions
type: Item
instance of: scholarly article
title: On differentiability and analyticity of positive definite functions
zbMATH Open document ID: 1215.42015
author: Jorge Buescu
author: A. C. Paixão
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-01-12T00:00:00Z
full work available at URL: http://hdl.handle.net/10400.21/674
review text: The paper deals with positive definite functions \(f:\mathbb{R} \to \mathbb{C}\), that is such that the inequality \(\sum_{j,k=1}^n f(x_j-x_k) \xi_j \overline{\xi_k} \geq 0\) holds for every choice of \(x_1,x_2,\ldots,x_n \in \mathbb{R}\) and \(\xi_1,\xi_2,\ldots,\xi_n \in \mathbb{C}.\) All results of the paper demonstrate the following effects: the global behavior of a smooth positive definite function is determined solely by the sequence of even-order derivatives at the origin. More precisely, the authors prove the following theorems. Theorem 1: Let \(f:\mathbb{R} \to \mathbb{C}\) be a positive definite function and suppose \(f\) is of class \(C^{2n}\) in a neighborhood of the origin for some positive integer \(n.\) If \(f^{(2m)}(0)=0\) for some positive integer \(m \leq n,\) then \(f\) is constant on \(\mathbb{R}\). Theorem 2: Let \(f:\mathbb{R} \to \mathbb{C}\) be a positive definite function and suppose \(f\) is of class \(C^{\infty}\) in a neighborhood of the origin. Then, if there exist positive constants \(M\) and \(D\) such that \[ 0\leq (-1)^n f^{(2n)}(0) \leq D\frac{(2n)!}{M^{2n}} \] for every non-negative integer \(n,\) we have: (i) \(f\) is analytic in \(\mathbb{R};\) (ii) Let \[ 1/h=\limsup \left({\frac{|f^{(2n)}(0)|}{(2n)!}}\right)^{\frac{1}{2n}}. \] There exist \(\alpha, \beta \in [h, +\infty]\) such that \(f\) holomorphically extends to the complex strip \(\{z\in \mathbb{C}: -\alpha < \mathrm{Im}(z) <\beta\},\) where \(\alpha\) and \(\beta\) are maximal with this property. Moreover, if \(h < \infty,\) the function \(f\) cannot be holomorphically extended to both the points \(z=ih\) and \(z=-ih\) simultaneously, implying in particular that \(h=\min\{\alpha,\beta\}\). The proof of these theorems are based on the fact that if a positive definite function \(f\) belongs to the class \(C^{2n}\) in a neighborhood of the origin for some positive integer \(n,\) then \(f \in C^{2n}(\mathbb{R})\) and for all integers \(0\leq m_1\leq n\), \(0\leq m_2\leq n\) an…
reviewed by: Olga M. Katkova
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 30B40
zbMATH DE Number: 5835304
zbMATH Keywords: positive definite function
zbMATH Keywords: smooth positive definite function
zbMATH Keywords: positive definite kernel
zbMATH Keywords: positive definite matrix
zbMATH Keywords: positive semidefinite matrix
zbMATH Keywords: real-analytic function
MaRDI profile type: Publication
OpenAlex ID: W2028287628
cites work: Theory of Reproducing Kernels
cites work: Q4459298
cites work: Q3228785
cites work: Positive definite matrices and differentiable reproducing kernel inequalities
cites work: Eigenvalue distribution of positive definite kernels on unbounded domains
cites work: A linear algebraic approach to holomorphic reproducing kernels in \(\mathbb C^n\)
cites work: Positive definite matrices and integral equations on unbounded domains
cites work: Quasianalyticity of positive definite continuous functions
cites work: Global regularity of positive define continous functions
cites work: On Infinitely Differentiable Positive Definite Functions
cites work: Q5581948
cites work: On Analytic Continuation to a Schlicht Function
cites work: Q3150781
cites work: On analytic characteristic functions
cites work: Analyticity and quasianalyticity of positive definite distributions
cites work: Q4291792
DOI: 10.1016/J.JMAA.2010.09.033
description: scientific article; zbMATH DE number 5835304