Uniform Boundedness of Sequence of Operators Associated with the Walsh System and Their Pointwise Convergence

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Abstract

Revisiting the main point of the almost everywhere convergence, it becomes clear that a weak (1,1)-type inequality must be established for the maximal operator corresponding to the sequence of operators. The better route to take in obtaining almost everywhere convergence is by using the uniform boundedness of the sequence of operator, instead of using the mentioned maximal type of inequality. In this paper it is proved that a sequence of operators, defined by matrix transforms of the Walsh–Fourier series, is convergent almost everywhere to the function f∈L1 if they are uniformly bounded from the dyadic Hardy space H1I to L1I. As a further matter, the characterization of the points are put forth where the sequence of the operators of the matrix transform is convergent.

Original languageEnglish
Article number24
JournalJournal of Fourier Analysis and Applications
Volume30
Issue number2
DOIs
Publication statusPublished - Apr 2024

Keywords

  • 42C10
  • Almost everywhere convergence
  • Boundedness of sequence of operators
  • Hardy spaces
  • Walsh system

ASJC Scopus subject areas

  • Analysis
  • General Mathematics
  • Applied Mathematics

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