An analog of Rudin's theorem for continuous radial positive definite functions of several variables
type: Item
instance of: scholarly article
title: An analog of Rudin's theorem for continuous radial positive definite functions of several variables
zbMATH Open document ID: 1312.26029
DOI: 10.1134/S0081543814020072
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-12-17T00:00:00Z
review text: A function \(f:{\mathbb{R}}^m\to {\mathbb{R}}\) is called radial if the value \(f(x)\) depends only on the norm of \(x\). Let \({\mathcal{G}}_m\) be the class of continuous radial functions with support in the unit ball and nonegative Fourier transform. The main result of the paper says that any \(\varphi \in {\mathcal{G}}_m\) can be represented as the sum \(\sum f_k\ast f_k\) of self convolutions of real-valued functions \(f_k\in L_2(\mathbb{R}^m)\) with supports in the ball of radius 1/2. This is a generalization of the result of \textit{W. Rudin} [Duke Math. J. 37, 49--53 (1970; Zbl 0194.36001)] from 1970.
reviewed by: Andrzej Nowak
Mathematics Subject Classification ID: 26B40
Mathematics Subject Classification ID: 26B35
zbMATH DE Number: 6381065
zbMATH Keywords: radial function
zbMATH Keywords: positive definite functions
zbMATH Keywords: Rudin's theorem
MaRDI profile type: Publication
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full work available at URL: s0081543814020072
OpenAlex ID: W2052851820
description: scientific article; zbMATH DE number 6381065