Herz spaces and restricted summability of Fourier transforms and Fourier series

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Abstract

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1, 1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σTθ f → f over a cone-like set a.e. for all f ∈ L1 (Rd). Moreover, σTθ f (x) converges to f (x) over a cone-like set at each Lebesgue point of f ∈ L1 (Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

Original languageEnglish
Pages (from-to)42-54
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume344
Issue number1
DOIs
Publication statusPublished - Aug 1 2008

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Keywords

  • Cone-like sets
  • Hardy-Littlewood maximal function
  • Herz spaces
  • Lebesgue points
  • Restricted convergence
  • Wiener amalgam spaces
  • θ-Summation of Fourier series

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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