Abstract
Let (X,F, μ) be a finite positive measure space and {ϕ j,k (x): j, k = 1, 2,...} be a double orthonormal system of real-valued functions on X. We investigate the pointwise convergence of the double orthogonal series (2.1) in Pringsheim’s sense and in the regular sense introduced by Hardy, as well as its Cesàro (C, 1, 1) summability and its strong Cesàro |C, 1, 1| summability. In our main theorem (Theorem 2 in Section 3 below) we extend a previous result of Borgen [2] from single to double orthogonal series. The key ingredient of our proof is the extension of the familiar Kronecker lemma from single to double sequences of numbers (see in [8, Theorem 1]. As an application of our Theorem 2, we are able to conclude the a.e. statistical convergence of the double orthogonal series (2.1) under a weaker condition than (2.5) in the Rademacher–Menshov theorem (see Theorem 3 in the last Section 6).
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Dedicated to the memory of Professor Károly Tandori, the founder of the “orthogonal school” at the Bolyai Institute, on the 10th anniversary of his death
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Móricz, F. Strong Cesàro |C, 1, 1| summability and statistical convergence of double orthogonal series. Anal Math 43, 103–116 (2017). https://doi.org/10.1007/s10476-017-0107-7
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DOI: https://doi.org/10.1007/s10476-017-0107-7
Key words and phrases
- double orthogonal series
- pointwise convergence in Pringsheim’s sense
- regular convergence
- strong Cesàro |C, 1, 1| summability
- statistical convergence
- Kronecker lemma for double sequences