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Monatshefte für Mathematik

, Volume 175, Issue 1, pp 143–160 | Cite as

Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

  • Ferenc WeiszEmail author
Article

Abstract

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \)-summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \)-means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\). Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\), where \(1<r\le \infty \). Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\).

Keywords

Wiener amalgam spaces Herz spaces Strong Hardy-Littlewood maximal function \(\theta \)-summability Lebesgue points. 

Mathematics Subject Classification (2000)

Primary 42B08 46E30 Secondary 42B30 42A38 

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of Numerical AnalysisEötvös L. UniversityBudapest Hungary

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