Lebesgue Points of Two-Dimensional Fourier Transforms and Strong Summability

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We introduce the concept of modified strong Lebesgue points and show that almost every point is a modified strong Lebesgue point of $$f$$f from the Wiener amalgam space W(L1,ℓ)(R2). A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function θ. Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f∈W(L1,ℓ)(R2) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of f∈W(Lp,ℓ)(R2), whenever 1<p<∞. As an application we generalize the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for f∈W(L1,ℓ)(R) and for strong θ-summability. Some special cases of the θ-summation are considered, such as the Weierstrass, Abel, Picar, Bessel, Fejér, de La Vallée-Poussin, Rogosinski and Riesz summations.

Original languageEnglish
Pages (from-to)885-914
Number of pages30
JournalJournal of Fourier Analysis and Applications
Volume21
Issue number4
DOIs
Publication statusPublished - Aug 25 2015

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Lebesgue Point
Summability
Fourier transform
Fourier transforms
Mercury amalgams
Summation
Wiener Amalgam Spaces
Friedrich Wilhelm Bessel
Converge
Generalise

Keywords

  • Fejér summability
  • Fourier transforms
  • Lebesgue points
  • Marcinkiewicz summability
  • Strong summability
  • θ-Summability

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

Cite this

Lebesgue Points of Two-Dimensional Fourier Transforms and Strong Summability. / Weisz, F.

In: Journal of Fourier Analysis and Applications, Vol. 21, No. 4, 25.08.2015, p. 885-914.

Research output: Contribution to journalArticle

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