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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Almost everywhere convergence of Vilenkin-Fourier series

Author: John Gosselin
Journal: Trans. Amer. Math. Soc. 185 (1973), 345-370
MSC: Primary 43A70; Secondary 42A20
MathSciNet review: 0352883
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Abstract: It is shown that the partial sums of Vilenkin-Fourier series of functions in ${L^q}(G),q < 1$, converge almost everywhere, where G is a zero-dimensional, compact abelian group which satisfies the second axiom of countability and for which the dual group X has a certain bounded subgroup structure. This result includes, as special cases, the Walsh-Paley group ${2^w}$, local rings of integers, and countable products of cyclic groups for which the orders are uniformly bounded.

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Keywords: Compact abelian group, character, Plancherel theorem, martingale maximal function, weak type inequalities
Article copyright: © Copyright 1973 American Mathematical Society