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Transactions of the American Mathematical Society

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On the a.e. convergence of the arithmetic means of double orthogonal series

Author: F. Móricz
Journal: Trans. Amer. Math. Soc. 297 (1986), 763-776
MSC: Primary 42B05; Secondary 42A24
MathSciNet review: 854098
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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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Keywords: Double orthogonal series, rectangular partial sums, $ (C,1,1)$, $ (C,1,0)$, $ (C,0,1)$-means, convergence in Pringsheim's sense, regular convergence, coefficient tests for a.e. summability
Article copyright: © Copyright 1986 American Mathematical Society