Annals of Functional Analysis

On pointwise inversion of the Fourier transform of $BV_{0}$ functions

Francisco J‎. ‎Mendoza Torres

Full-text: Open access

Abstract

‎Using a Riemann-Lebesgue lemma for the Fourier transform over the class of‎ ‎bounded variation functions that vanish at infinity‎, ‎we prove the‎ ‎Dirichlet--Jordan theorem for functions on this class‎. ‎Our proof is in the‎ ‎Henstock--Kurzweil integral context and is different to that of‎ ‎Riesz-Livingston [Amer‎. ‎Math‎. ‎Monthly 62 (1955)‎, ‎434--437]‎. ‎As consequence‎, ‎we obtain the Dirichlet--Jordan theorem‎ ‎for functions in the intersection of the spaces of bounded variation‎ ‎functions and of Henstock--Kurzweil integrable functions‎. ‎In this‎ ‎intersection there exist functions in $L^{2}(\mathbb{R})\backslash L(\mathbb{%‎ ‎R}).$‎

Article information

Source
Ann. Funct. Anal., Volume 1, Number 2 (2010), 112-120 .

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1399900593

Digital Object Identifier
doi:10.15352/afa/1399900593

Mathematical Reviews number (MathSciNet)
MR2772044

Zentralblatt MATH identifier
1217.42016

Subjects
Primary: 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 26A39: Denjoy and Perron integrals, other special integrals

Keywords
Fourier transform ‎Henstock--Kurzweil integral ‎Dirichlet-Jordan‎ ‎theorem

Citation

‎Mendoza Torres, Francisco J‎. On pointwise inversion of the Fourier transform of $BV_{0}$ functions. Ann. Funct. Anal. 1 (2010), no. 2, 112--120. doi:10.15352/afa/1399900593. https://projecteuclid.org/euclid.afa/1399900593


Export citation