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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A few remarks on Riesz summability of orthogonal series
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by PawełJ. Szabłowski
Proc. Amer. Math. Soc. 113 (1991), 65-75
DOI: https://doi.org/10.1090/S0002-9939-1991-1072349-8

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 $i = 1,2, \ldots ,j = 1, \ldots ,k$, \[ {\ln ^{(1)}}i = {\ln _2}i,\quad {\ln ^{(j)}}i = {\ln _2}(\max (1,{\ln ^{(j - 1)}}i)).\]
References
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  • B. S. Kashin and A. A. Saakyan, Ortogonal′nye ryady, “Nauka”, Moscow, 1984 (Russian). MR 779286
  • Yasuo Okuyama and Tamotsu Tsuchikura, On the absolute Riesz summability of orthogonal series, Anal. Math. 7 (1981), no. 3, 199–208 (English, with Russian summary). MR 635485, DOI 10.1007/BF01908522
  • P. J. Szabłowski, Stochastic approximation with dependent disturbances. I, Comput. Math. Appl. 13 (1987), no. 12, 951–972. MR 898943, DOI 10.1016/0898-1221(87)90067-8
  • —, Application of generalized laws of large numbers to the proper choice of amplification coefficients in stochastic approximation with correlated disturbances, Proc. of 3rd Kinston Conf., Lecture Notes in Pure and Appl. Math., vol. 44, Marcel Dekker, New York, 1978, pp. 223-243. N. N. Vorobiov, Teoria riadov, izd. Nauka, Moskva, 1986. (Russian)
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Bibliographic Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 113 (1991), 65-75
  • MSC: Primary 42C15
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1072349-8
  • MathSciNet review: 1072349