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A few remarks on Riesz summability of orthogonal series

Author: PawełJ. Szabłowski
Journal: Proc. Amer. Math. Soc. 113 (1991), 65-75
MSC: Primary 42C15
MathSciNet review: 1072349
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Abstract: We study convergence behavior of some sequences and series related to a given orthogonal series. Following the developed technique we define (in terms of fourth mixed moments only) a class of orthonormal functions $ {\left\{ {{X_i}} \right\}_{i \geq 1}}$ such that the condition: $ \exists k \in \mathbb{N}\sum\nolimits_{i \geq 1} {\mu _i^2} {\left( {{{\ln }^{\left( k \right)}}i} \right)^2} < \infty $ implies almost everywhere convergence of the series $ \sum\nolimits_{i \geq 1} {{\mu _i}{X_i}} $, here for every $ i = 1,2, \ldots ,j = 1, \ldots ,k$,

$\displaystyle {\ln ^{(1)}}i = {\ln _2}i,\quad {\ln ^{(j)}}i = {\ln _2}(\max (1,{\ln ^{(j - 1)}}i)).$

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Keywords: Orthognal series, almost everywhere convergence, laws of large numbers, Riesz summation
Article copyright: © Copyright 1991 American Mathematical Society

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