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Journal of Fourier Analysis and Applications

, Volume 18, Issue 2, pp 266–286 | Cite as

Almost Everywhere Convergent Fourier Series

  • M. J. Carro
  • M. Mastyło
  • L. Rodríguez-Piazza
Article

Abstract

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.

Keywords

Fourier series Almost everywhere convergence Lorentz spaces Banach envelope Rearrangement invariant spaces 

Mathematics Subject Classification (2000)

47A30 47A63 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • M. J. Carro
    • 1
  • M. Mastyło
    • 2
  • L. Rodríguez-Piazza
    • 3
  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz University Poznań, and Institute of Mathematics Polish Academy of Sciences (Poznań branch)PoznańPoland
  3. 3.Faculty of MathematicsUniversity of SevillaSevillaSpain

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