On pointwise inversion of the Fourier transform of \(\mathrm{BV}_0\) functions
type: Item
instance of: scholarly article
title: On pointwise inversion of the Fourier transform of \(\mathrm{BV}_0\) functions
zbMATH Open document ID: 1217.42016
published in: Annals of Functional Analysis
publication date: 2011-05-02T00:00:00Z
full work available at URL: https://eudml.org/doc/233568
full work available at URL: AFA-tex v1 n2 a12.pdf
full work available at URL: http://projecteuclid.org/euclid.afa/1399900593
review text: The Dirichlet-Jordan theorem solves the pointwise inversion of the Fourier transform for Lebesgue integrable and bounded variation functions on \(\mathbb R\). Using a Riemann-Lebesgue lemma for the Fourier transform over the class of bounded variation functions that vanish at infinity, the Dirichlet-Jordan theorem is proved for functions in this class. The proof given in this paper is in the Henstock-Kurzweil integral context and is different to that of Riesz-Livingston. As a consequence, the Dirichlet-Jordan theorem for functions in the intersection of the spaces of bounded variation functions and of Henstock-Kurzweil integrable functions is obtained. In this intersection there exist functions that are square integrable, but not Lebesgue integrable.
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 26A39
zbMATH DE Number: 5883044
zbMATH Keywords: Fourier transform
zbMATH Keywords: Henstock-Kurzweil integral
zbMATH Keywords: Dirichlet-Jordan theorem
author: Francisco J. Mendoza Torres
MaRDI profile type: Publication
OpenAlex ID: W2061975282
DOI: 10.15352/AFA/1399900593
description: scientific article; zbMATH DE number 5883044