1-Summability of d-Dimensional Fourier Transforms

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Abstract

A general summability method of more-dimensional Fourier transforms is given with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the ℓ1-θ-means of a tempered distribution is bounded from Hp(d) to Lp(d) for all d/(d+α)<p≤∞ and, consequently, is of weak type (1,1), where 0<α≤1 depends only on θ. As a consequence we obtain a generalization of the one-dimensional summability result due to Lebesgue, more exactly, the ℓ1-θ-means of a function f∈L1(d) converge a. e. to f. Moreover, we prove that the ℓ1-θ-means are uniformly bounded on the spaces Hp(d), and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the ℓ1-θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.

Original languageEnglish
Pages (from-to)421-452
Number of pages32
JournalConstructive Approximation
Volume34
Issue number3
DOIs
Publication statusPublished - Dec 1 2011

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Keywords

  • Fourier transforms
  • Hardy spaces
  • Interpolation
  • p-atom
  • ℓ-θ-summation

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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