Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series

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Abstract

The martingale Hardy space Hp([0,1)2) and the classical Hardy space Hp(double-struck T sign2) are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from Hp to Lp (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a Marcinkiewicz-Zygmund type inequality is obtained for BMO spaces.

Original languageEnglish
Pages (from-to)173-194
Number of pages22
JournalStudia Mathematica
Volume117
Issue number2
DOIs
Publication statusPublished - 1996

Keywords

  • Atomic decomposition
  • Hardy-Littlewood inequality
  • Martingale and classical Hardy spaces
  • P-atom
  • Walsh functions

ASJC Scopus subject areas

  • Mathematics(all)

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