On the Hilbert transform
type: Item
instance of: scholarly article
title: On the Hilbert transform
zbMATH Open document ID: 0628.44011
DOI: 10.1007/BF02202571
author: R. I. Gurielashvili
published in: Analysis Mathematica
publication date: 1987-01-01T00:00:00Z
review text: The Hilbert transform of f is given by the formula \(Hf(x)=(1/\pi)\int^{\infty}_{-\infty}(x-t)^{-1}f(t)dt,\) where the integral is taken in the principal value sense. Let \(L^*\) be the collection of all function f such that \((1+| t|)^{-1}f(t)\) is integrable on (-\(\infty,\infty)\), and let \(L^ p_{\alpha}(R)\) be the class of function f for which \(\| f\|_{p,\alpha}=(\int_{R}| f(t)|^ p| t|^{\alpha}dt)^{1/p}<\infty,\) where \(p>0\), \(R=(-\infty,\infty)\) and \(\alpha\in R\). The author obtains some necessary and sufficient conditions for the function \(f\in L^ p_{\alpha}(R)\cap L^*\) to have its Hilbert transform Hf in \(L^ p_{\alpha}(R)\).
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 4021860
zbMATH Keywords: integrability of the pth degree
zbMATH Keywords: Hilbert transform
reviewed by: Y. Tanno
MaRDI profile type: Publication
cites work: FINITENESS OF $ \textit{Ø}$ OVER TOTALLY REAL FIELDS
cites work: Q4065523
cites work: On Conjugate Functions
cites work: Some more theorems concerning Fourier series and Fourier power series
description: scientific article; zbMATH DE number 4021860