Abstract
For a Haar-system series we prove that if the lower bound of the (C, 1) means of the series is larger than — ∞ on a set E of positive measure, then the series converges to a finite function almost everywhere on E; from this it follows that Haar-system series are not summable by the (C, 1) method to + ∞ on sets of positive measure.
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S. Kaczmarz and H. Steinhaus, Theory of Orthogonal Series [Russian translation], Moscow (1958).
A. A. Talalyan and F. G. Arutyunyan, “On the convergence to Haar-system series,” Matem. Sb.,66, No. 2, 240–247 (1965).
F. G. Arutyunyan, “On Haar-system series,” Dokl. Akad. Nauk Arm. SSR,42, No. 3, 134–140 (1966).
R. F. Gundy, “Martingale theory and pointwise convergence of certain orthogonal series,” Trans. Amer. Math. Soc.,124, No. 2, 228–248 (1966).
V. A. Skvortsov, “Differentiation with respect to nets and Haar series,” Matem. Zametki,4, No. 1, 33–40 (1968).
P. L. Ul'yanov, “On Haar-system series with monotonic coefficients,” Izv. Akad. Nauk SSSR, Ser. Matem.,28, No. 4, 948 (1964).
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Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 393–404, March, 1974.
In conclusion, I wish to express my thanks to F. G. Arutyunyan for his statement of the problem and subsequent discussion.
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Shaginyan, L.A. Summability of series with respect to a Haar system by the (C, 1) method. Mathematical Notes of the Academy of Sciences of the USSR 15, 226–233 (1974). https://doi.org/10.1007/BF01438375
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DOI: https://doi.org/10.1007/BF01438375