On rearrangements of Vilenkin-Fourier series which preserve almost everywhere convergence

Gosselin, J. A.; Young, W. S.

doi:10.1090/S0002-9947-1975-0399756-6

On rearrangements of Vilenkin-Fourier series which preserve almost everywhere convergence
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by J. A. Gosselin and W. S. Young

Trans. Amer. Math. Soc. 209 (1975), 157-174

DOI: https://doi.org/10.1090/S0002-9947-1975-0399756-6

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Abstract:

It is known that the partial sums of Vilenkin-Fourier series of ${L^q}$ functions $(q > 1)$ converge a.e. In this paper we establish the ${L^2}$ result for a class of rearrangements of the Vilenkin-Fourier series, and the ${L^q}$ result $(1 < q < 2)$ for a subclass of rearrangements. In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. Balashov. The ${L^2}$ result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.

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