On a problem of H. Shapiro
type: Item
instance of: scholarly article
title: On a problem of H. Shapiro
zbMATH Open document ID: 1048.31002
author: I. V. Ostrovskii
author: Alexander Ulanovskii
published in: Journal of Approximation Theory
publication date: 2004-08-10T00:00:00Z
full work available at URL: http://hdl.handle.net/11693/11277
review text: Let \(\mu\) be a real measure on the real line. Assume that its Poisson integral \(M(z)\), \(z\in \mathbb C\setminus \mathbb R\) satisfies \[ | M(x+iy)| \leq Ae^{-cy^\alpha},\;\;y\to \infty, \] for some constants \(A,c>0\) and \(0<\alpha\leq 1\). It is known that this estimate is equivalent (when \(\mu\) is absolutely continuous) with the condition that the support of the Fourier transform of \(\mu\) is disjoint from the interval \((-c,c)\). The authors show that for \(1/2<\alpha\leq 1\), the measure \(\mu\) must have many sign changes on both positive and negative rays. For \(0<\alpha\leq 1/2\), this is true for at least one of the rays, and not always true for both rays. The authors provide examples that show that their asymptotic bounds for the number of sign changes are sharp.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 31A10
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 2086001
zbMATH Keywords: Poisson integral
zbMATH Keywords: sign changes
zbMATH Keywords: asymptotic bounds
MaRDI profile type: Publication
OpenAlex ID: W2079968993
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cites work: Q3735790
DOI: 10.1016/J.JAT.2003.12.003
description: scientific article; zbMATH DE number 2086001