Remarks on a theorem by N. Yu. Antonov
Volume 158 / 2003
Studia Mathematica 158 (2003), 79-97
MSC: 42A20, 42B25.
DOI: 10.4064/sm158-1-7
Abstract
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.
