On a theorem of Akhiezer
type: Item
instance of: scholarly article
title: On a theorem of Akhiezer
zbMATH Open document ID: 0792.44006
author: Elizabeth Kochneff
author: Ruby Tan
author: Yoram Sagher
published in: Illinois Journal of Mathematics
publication date: 1994-08-07T00:00:00Z
review text: The authors extend Akhiezer's theorem about the relation between the Hilbert transforms of the function \(f(x) \in L_ 2 (R)\) and the function \(f(x)/(x-\gamma) \in L_ 2 (R)\), where \(\gamma\) is a constant on the functions from the weighted space \(L_ 1 (R,dt/(1+t^ 2))\). They assume also that \((f(x)-\alpha)/(t-\gamma)\) is a locally integrable function. Akhiezer's proof depends on calculations of Fourier transforms, using complex methods and therefore it was difficult to generalize this theorem for \(L_ p\)-space with \(p \neq 2\). The authors overcome these problems and provide the calculation of new examples of the Hilbert transforms of some functions. Considering rational weights for \(L_ 1\)-functions they obtain new interesting examples Akhiezer's type relations for the Hilbert transform.
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 561225
zbMATH Keywords: Poisson kernel
zbMATH Keywords: Hilbert transforms
reviewed by: S. Yakubovich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 561225