An analogue of a theorem of Ferenc Lukács for double Walsh-Fourier series
type: Item
instance of: scholarly article
title: An analogue of a theorem of Ferenc Lukács for double Walsh-Fourier series
zbMATH Open document ID: 1006.42024
DOI: 10.1023/A:1015653309520
published in: Acta Mathematica Hungarica
publication date: 2003-02-18T00:00:00Z
review text: A theorem of Ferenc Lukács states that if a periodic function \(f\) is integrable in the Lebesgue's sense and has discontinuity of first kind at some point \(\alpha\), then the \(n\)th partial sum of the conjugate series to its trigonometric Fourier series at \(\alpha\) divided by \(\log n\) converges to \(\pi^{-1}\{f(\alpha-0)-f(\alpha+0)\}\) as \(n\to\infty\). An analogue of this theorem for Walsh-Fourier series is proved by \textit{R. Riad} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 30, 69-76 (1987; Zbl 0644.42025)]. The main aim of this paper is to extend the latter result from single to double Walsh-Fourier series. Given a number \(\alpha\in[0,1)\), consider its binary representation \(\alpha=\sum_{k=0}^{\infty}\alpha_k2^{-k-1}\). The \(n\)th variation \(V_n(\alpha)\) of \(\alpha\) is defined as \(V_n(\alpha):=|\alpha_0|+\sum_{k=0}^{n- }|\alpha_{k+1}-\alpha_k|\). Define the \(2^n\)th conjugate Walsh-Dirichlet kernel \(\tilde D_{2^n}^{\alpha} :=\sum_{k=0}^{n-1}(-1)^{r_k}D_{2^k}\), where \(r_k\) is the \(k\)th Rademacher function. The \(2^n\)th partial sum of the conjugate series is the dyadic convolution of the function \(f\) and \(\tilde D_{2^n}^{\alpha}\). Set \(\lambda_n(\alpha):=(\tilde S_{2^n}^{\alpha}\chi_{[0,\alpha)})(\alpha)\), where \(\chi_I\) denotes the characteristic function of the interval \(I\). The main theorem of the paper is as follows. \noindent Theorem. Let \(\alpha,\beta\in (0,1)\) be such that \(\lim _m(\alpha)=\lim V_n(\beta) =\infty\), and \(f\in L^1[0,1)^2\) such that \[ \int_0^1 |f(x,y)|dy , \int_0^1 |f(x,y)|dx \leq B \] for \(x,y\in (0,1)\), where \(B\) is some positive constant. If \(f\) has discontinuity of first kind at the point \((\alpha,\beta)\), then \[ \lim_{m,n}\frac{\tilde S_{2^m,2^n}^{\alpha,\beta}(\alpha,\beta)}{\lambda_m(\alpha)\lambda_n(\beta)} = f(\alpha+0, \beta+0)- f(\alpha-0, \beta+0)- f(\alpha+0, \beta-0)+ f(\alpha-0, \beta-0). \] The authors also consider functions of two variables which are of bounded vari…
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 26B30
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 1870473
zbMATH Keywords: Walsh-Paley system
zbMATH Keywords: Walsh series
zbMATH Keywords: conjugate Walsh series
zbMATH Keywords: rectangular partial sum
zbMATH Keywords: bounded variation over a rectangle in the sense of Hardy and Krause
zbMATH Keywords: discontinuity of first kind
zbMATH Keywords: sector limits of a function of two variables
author: Ferenc Móricz
author: William R. Wade
MaRDI profile type: Publication
full work available at URL: a:1015653309520
OpenAlex ID: W1898379357
description: scientific article; zbMATH DE number 1870473