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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the a.e. convergence of the arithmetic means of double orthogonal series
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Trans. Amer. Math. Soc. 297 (1986), 763-776 Request permission

Abstract:

The extension of the coefficient test of Menšov and Kaczmarz ensuring the a.e. $(C,1,1)$-summability of double orthogonal series has been stated by two authors. Unfortunately, their proofs turned out to be deficient. Now we present a general theory, in the framework of which a complete proof of this test can also be obtained. Besides, we extend the relevant theorems of Kolmogorov and Kaczmarz from single orthogonal series to double ones, establishing the a.e. equiconvergence of the lacunary subsequences of the rectangular partial sums and of the entire sequence of the arithmetic means. The corresponding tests ensuring the a.e. $(C,1,0)$ and $(C,0,1)$-summability are also treated.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 297 (1986), 763-776
  • MSC: Primary 42B05; Secondary 42A24
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0854098-4
  • MathSciNet review: 854098