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Andrienko, Vitalii A.

To the question on the almost everywhere convergence of the Riesz means of double orthogonal series. (English) Zbl 1256.42043

J. Math. Sci., New York 183, No. 6, 749-761 (2012); translation from Ukr. Mat. Visn. 9, No. 1, 1-17 (2012).
Summary: Sufficient conditions of the classical type ensuring the almost everywhere (a.e.) convergence of the nonnegative-order Riesz means of double orthogonal series are indicated. Analogies of the onedimensional results of A. Kolmogoroff [Fund. Math. 5, 96–97 (1924; JFM 50.0206.04)] and St. Kaczmarz [Math Z. 26, 99–105 (1927; JFM 53.0267.01)] and A. Zygmund [Fundamenta 10, 356–362 (1927; JFM 53.0267.04)] have been obtained for the Cesaro means and those of A. Zygmund [Bulletin Polon. Acad. Sci., Ser. A 1927, 295–308 (1927; JFM 53.0266.01)] for the Riesz means. These analogies establish the a.e. equiconvergence of the lacunary subsequences of rectangular partial sums and of the entire sequence of Riesz means, generalize the corresponding results of F. Móricz [Trans. Am. Math. Soc. 297, 763–776 (1986; Zbl 0605.42024)] for the Cesaro a.e. summability by \((C, 1, 1)\), \((C, 1, 0)\), and \((C, 0, 1)\) methods of double orthogonal series, and were announced earlier without proofs in the author’s work [Dokl. Akad. Nauk Ukr. SSR, Ser. A 1989, No. 2, 3–5 (1989; Zbl 0698.40005)].

MSC:

42C15 General harmonic expansions, frames

Keywords:

double orthogonal series; Riesz means

Citations:

Zbl 0605.42024; Zbl 0698.40005; JFM 50.0206.04; JFM 53.0267.01; JFM 53.0267.04; JFM 53.0266.01
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Full Text: DOI

References:

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[3] V. Andrienko, ”Summability of double orthogonal series by the Riesz methods,” Dokl. AN UkrSSR. Ser. A. Fiz.-Mat. Tekh. Nau., No. 2, 3–5 (1989). · Zbl 0698.40005
[4] V. Andrienko, ”On the a.e. convergence of the Riesz means of double orthogonal series,” Ukr. Mat. Zh., 51, 867–880 (1999). · Zbl 0954.42017 · doi:10.1007/BF02529550
[5] S. Kaczmarz, ”Über die Reihen von allgemeinen Orthogonalfunktionen,” Math. Ann., 96, 148–151 (1925). · JFM 52.0276.01 · doi:10.1007/BF01209157
[6] S. Kaczmarz, ”Über die Summierbarkeit der Orthogonalreihen,” Math. Zh., 26, 99–105 (1927). · JFM 53.0267.01 · doi:10.1007/BF01475443
[7] A. Kolmogoroff, ”Une contribution à l’etude de la convergence des séries de Fourier,” Fund. Math., 5, 96–97 (1924). · JFM 50.0206.04
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[9] F. Móricz, ”On the a.e. convergence of the arithmetic means of double orthogonal series,” Trans. Amer. Math. Soc., 297, 763–776 (1986). · Zbl 0605.42024
[10] F. Móricz, ”On the convergence in a restricted sense of multiple series,” Anal. Math., 5, 135–147 (1979). · Zbl 0428.40001 · doi:10.1007/BF02059384
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[12] A. Zygmund, ”Sur l’application de la première moyenne arithmétique dans la théorie des séries orthogonales,” Fund. Math., 10, 356–362 (1927). · JFM 53.0267.04
[13] A. Zygmund, ”Sur la sommation des séries de fonctions orthogonales,” Bull. Int. Acad. Polon. Sci. Lett. (Cracovie). Sér. A, No. 6, 295–308 (1927). · JFM 53.0266.01
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