Remarks on a theorem by N. Yu. Antonov

Tom 158 / 2003

Per Sjölin, Fernando Soria Studia Mathematica 158 (2003), 79-97 MSC: 42A20, 42B25. DOI: 10.4064/sm158-1-7


We extend some results of N. Yu. Antonov on convergence of Fourier series to more general settings. One special feature of our work is that we do not assume smoothness for the kernels in our hypotheses. This has interesting applications to convergence with respect to general orthonormal systems, like the Walsh–Fourier system, for which we prove a.e. convergence in the class $L\mathop {\rm log}\nolimits L \mathop {\rm log}\nolimits \mathop {\rm log}\nolimits \mathop {\rm log}\nolimits L$. Other applications are given in the theory of differentiation of integrals.


  • Per SjölinDepartment of Mathematics
    Royal Institute of Technology
    S-100 44 Stockholm, Sweden
  • Fernando SoriaDepartment of Mathematics, C-XV
    Universidad Autónoma de Madrid
    E-28049 Madrid, Spain

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