Abel derivative and Abel continuity
type: Item
instance of: scholarly article
title: Abel derivative and Abel continuity
zbMATH Open document ID: 1273.26005
author: Yanyan Li
published in: Real Analysis Exchange
publication date: 2013-05-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rae/1367265640
review text: The concept of an Abel derivative is given in the paper of \textit{S. Verblunsky} [Proc. Lond. Math. Soc., II. Ser. 34, 441--456 (1932; Zbl 0006.05302)], and the book of \textit{A. Zygmund} [Trigonometric series. Volumes I and II combined. With a foreword by Robert Fefferman. 3rd ed. Cambridge: Cambridge University Press (2002; Zbl 1084.42003)], but the first definition was that of the second order Abel derivative by \textit{S. J. Taylor} [Q. J. Math., Oxf. II. Ser. 6, 255--274 (1955; Zbl 0066.30002)]. Since then little was done on these derivatives although the second author did some work in [J. Lond. Math. Soc., II. Ser. 17, 87--96 (1978; Zbl 0385.42003); J. Lond. Math. Soc., II. Ser. 20, 319--326 (1979; Zbl 0423.42006)]. Finally, Abel derivatives of all orders were defined in a recent book, ``Higher order derivatives'', by the second author [Boca Raton, FL: CRC Press (2012; Zbl 1247.26004)]. If \(f(r,x) ={{1}\over{2}}a_0 + \sum_{n=1}^{\infty}(a_n\cos nx + b_n\sin nx)r^n\) then the upper and lower Abel derivates of \(f\) at \(x\) of order \(k\) are defined by \( \overline {AD}_kf(x)= \limsup_{r\to 1-}\frac{\partial^kf}{\partial x^k}(r,x),\;\underline {AD}_kf(x)= \liminf_{r\to 1-}\frac{\partial^kf}{\partial x^k}(r,x) \); in particular, the Abel derivative at \(x\) of order \(0\) is just the Abel sum at \(x\) of the Fourier series of \(f\). In the aforementioned book the basic properties of these derivatives were obtained and relations to the symmetric de la Vallée Poussin derivatives, also called the symmetric Peano derivatives, explored. The current paper takes up the detailed examination of properties of these derivatives, especially of the first order derivative which for Lebesgue integrable functions is a generalisation of the ordinary first symmetric derivative. In this case the derivative is shown to be a Baire-1 function and a monotonicity theorem is proved. Extensions of the results to the higher order derivatives present difficulties although the second au…
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 42A24
zbMATH DE Number: 6168745
zbMATH Keywords: Abel summability
zbMATH Keywords: Fourier series
zbMATH Keywords: Abel continuity
zbMATH Keywords: Abel derivative
zbMATH Keywords: monotonicity
reviewed by: P. S. Bullen
MaRDI profile type: Publication
OpenAlex ID: W1582062454
cites work: A general integral
cites work: On the Abel Summability of Trigonometric Series
cites work: On the Abel Limit of the Terms of Trigonometric Series
cites work: Q5536399
cites work: AN INTEGRAL OF PERRON'S TYPE DEFINED WITH THE HELP OF TRIGONOMETRIC SERIES
cites work: Q4308746
cites work: On the Theory of Trigonometric Series (I)
cites work: On multipliers preserving convergence of trigonometric series almost everywhere
DOI: 10.14321/REALANALEXCH.38.1.0063
description: scientific article; zbMATH DE number 6168745