Almost everywhere convergence of cone-like restricted two-dimensional Fejér means with respect to Vilenkin-like systems

Nagy, K.

doi:10.1090/S1061-0022-2014-01309-X

Almost everywhere convergence of cone-like restricted two-dimensional Fejér means with respect to Vilenkin-like systems
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by K. Nagy

St. Petersburg Math. J. 25 (2014), 605-614

DOI: https://doi.org/10.1090/S1061-0022-2014-01309-X

Published electronically: June 5, 2014

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Abstract:

For the two-dimensional Walsh system, Gát and Weisz proved the a.e. convergence of the Fejér means $\sigma _n f$ of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, $\beta ^{-1}\leq n_1/n_2\leq \beta$, with some fixed parameter $\beta \geq 1$. The result of Gát and Weisz was generalized by Gát and the author in the way that the indices are inside a cone-like set.

In the present paper, the a.e. convergence is proved for the Fejér means of integrable functions with respect to two-dimensional Vilenkin-like systems provided that the set of indeces is in a cone-like set. That is, the result of Gát and the author is generalized to a general orthonormal system, which contains as special cases the Walsh system, the Vilenkin system, the character system of the group of 2-adic integers, the UDMD system, and the representative product system of CTD (compact totally disconnected) groups.

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