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A Tauberian theorem for Vilenkin series

Author: W. R. Wade
Journal: Proc. Amer. Math. Soc. 131 (2003), 2877-2881
MSC (2000): Primary 42C10, 43A75
Published electronically: February 28, 2003
MathSciNet review: 1974345
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Abstract: There are a number of papers in the literature which contain Cesàro analogues of results already known for martingale sums of Vilenkin-Fourier series. We show that for Vilenkin systems of bounded type, these are not merely analogues but actually generalizations. Indeed, we prove that convergence of the Cesàro means of a Vilenkin series $S$ implies convergence of martingale partial sums of $S$ itself.

References [Enhancements On Off] (What's this?)

  • 1. J.E. Daly and K.L. Phillips, A note on $H^{1}$multipliers for locally compact Vilenkin groups, Canad. Math. Bull. 41 (1998), 392-397. MR 99j:43005
  • 2. W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, Inc., 1987. MR 88k:00002
  • 3. L.A. Shaginyan, On definite limits and sets of limit functions of Walsh series, Mat. Sbornik 95 (1974), 263-271.
  • 4. N. Ya. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk. SSSR, Ser. Mat. 11 (1947), 363-400. MR 9:224h

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Additional Information

W. R. Wade
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996

Received by editor(s): April 10, 2002
Published electronically: February 28, 2003
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society

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