Functions of bounded variation and their Fourier transforms
type: Item
instance of: scholarly article
title: Functions of bounded variation and their Fourier transforms
zbMATH Open document ID: 1418.42001
DOI: 10.1007/978-3-030-04429-9
author: Elijah Liflyand
published in: Applied and Numerical Harmonic Analysis
publication date: 2019-02-15T00:00:00Z
review text: Fourier series and transform have been extensively studied for over hundred years. Convergence of Fourier series and integrals of different classes of functions, including functions of bounded variations, has also been studied for many years. So, a question naturally arises: Is there a need for a book on the Fourier transform of functions of bounded variations? \par The author begins by answering this question and explaining that his book may be considered as a second volume of a relatively recently published book [\textit{J. Appell} et al., Bounded variation and around. Berlin: de Gruyter (2014; Zbl 1282.26001)]. Since the publication of the latter some new results, as well as, different approaches to old problems, including generalizations to multi-dimensions, have appeared which necessitated compiling them in a book format. \par The book comprises two parts: Part I, One-dimensional case, and Part II, Multi-dimensional case, they are preceded by an introduction and followed by an afterward and the bibliography. Part I is made up of Chapters 1--4 and Part II of Chapters 5--9. In the introduction, the author states that instead of studying the Fourier transform of functions of bounded variations, the emphasis will be on their cosine and sine transforms given by \[ \hat{f}_c (x)=\int_0^\infty f(t) \cos (xt) dt, \] and \[ \hat{f}_s (x)=\int_0^\infty f(t) \sin (xt) dt, \] which may be combined in a single transform of the form \[ \hat{f}_\gamma (x)=\int_0^\infty f(t) \cos (xt-\pi \gamma /2) dt, \] where $\gamma = 0 \text{ or } 1$. \par Also reviewed in the introduction are some tools that will be used in later chapters, such as the Hilbert transform and the so-called balance operator defined by \[ B_{\phi} g(x)=\frac{1}{x^2}\int_0^\infty g(t/x) \phi (t)dt, \] which for $\phi(t)=\sin t $, yields $ \hat{g}_s (x)=x B_{\sin x}g(x). $ \par One of the author's goals is to extend some of the one-dimensional results to higher dimensions. Because of the variety of notions and d…
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 42-01
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 26A45
zbMATH DE Number: 7024210
zbMATH Keywords: Fourier series
zbMATH Keywords: Fourier transform
zbMATH Keywords: functions of bounded variation
zbMATH Keywords: absolutely continuous functions
zbMATH Keywords: convex functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-04429-9
OpenAlex ID: W2945127873
description: scientific article; zbMATH DE number 7024210