Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series

György Gát, Ushangi Goginava

Research output: Contribution to journalArticlepeer-review

Abstract

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh- Fourier series is of type (H#,L1) provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in H#, so by the well-known density argument we have that t2n,2m f (x1, x2) → f (x1, x2) a. e. as m, n → ∞ for all f ∈ H#(⊃ L log+ L). We also prove the sharpness of this result. Namely, For all measurable function δ: [0,+∞) → [0,+∞), limt→∞ δ(t) = 0 we have a function f such as f ∈ Llog+ Lδ(L) and the two-dimensional Nörlund logarithmic means do not converge to f a.e. (in the Pringsheim sense) on I2.

Original languageEnglish
Pages (from-to)447-464
Number of pages18
JournalReal Analysis Exchange
Volume31
Issue number2
DOIs
Publication statusPublished - 2006
Externally publishedYes

Keywords

  • A. e. convergence and divergence
  • Double Walsh-Fourier series
  • Logarithmic means

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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