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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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


Author: K. Nagy
Original publication: Algebra i Analiz, tom 25 (2013), nomer 4.
Journal: St. Petersburg Math. J. 25 (2014), 605-614
MSC (2010): Primary 42C10, 43A75, 40G05
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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Additional Information

K. Nagy
Affiliation: Institute of Mathematics and Computer Sciences, College of Nyíregyháza, P.O. Box 166, Nyíregyháza, H-4400, Hungary
Email: nkaroly@nyf.hu

DOI: http://dx.doi.org/10.1090/S1061-0022-2014-01309-X
PII: S 1061-0022(2014)01309-X
Keywords: Vilenkin group, Vilenkin system, pointwise convergence, Fej\'er means, orthonormal systems, two-dimensional Fourier series, compact totally disconnected group
Received by editor(s): June 13, 2012
Published electronically: June 5, 2014
Additional Notes: The author was supported by the project TÁMOP-4.2.2.A-11/1/KONV-2012-0051
Article copyright: © Copyright 2014 American Mathematical Society