On the a.e. convergence of the arithmetic means of double orthogonal series
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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.References
-
P. R. Agnew, On double orthogonal series, Proc. London Math. Soc. (2) 33 (1932), 420-434.
- G. Alexits, Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris, 1961. Translated from the German by I. Földer. MR 0218827
- L. Csernyák, Bemerkung zur Arbeit von V. S. Fedulov “Über die Summierbarkeit der doppelten Orthogonalreihen”, Publ. Math. Debrecen 15 (1968), 95–98 (German). MR 238027
- V. S. Fedulov, On $(C, 1, 1)$-summability of a double orthogonal series, Ukrain. Mat. Ž. 7 (1955), 433–442 (Russian). MR 0077673 G. H. Hardy, On the convergence of certain multiple series, Proc. Cambridge Philos. Soc. 19 (1916-1919), 86-95.
- Stefan Kaczmarz, Über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 96 (1927), no. 1, 148–151 (German). MR 1512309, DOI 10.1007/BF01209157
- Stefan Kaczmarz, Über die Summierbarkeit der Orthogonalreihen, Math. Z. 26 (1927), no. 1, 99–105 (German). MR 1544844, DOI 10.1007/BF01475443 A. N. Kolmogoroff, Une contribution à l’étude de la convergence des séries de Fourier, Fund. Math. 5 (1924), 96-97. D. E. Menchoff, Sur les séries de fonctions orthogonales. II, Fund. Math. 8 (1926), 56-108.
- F. Móricz, Moment inequalities and the strong laws of large numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 4, 299–314. MR 407950, DOI 10.1007/BF00532956
- F. Móricz, On the convergence in a restricted sense of multiple series, Anal. Math. 5 (1979), no. 2, 135–147 (English, with Russian summary). MR 539321, DOI 10.1007/BF02059384
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
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