Geodesic
On the limits of indetermination and on the set of limit functions of series in the Walsh system
Sbornik. Mathematics, Tome 24 (1974) no. 2, pp. 257-265 
L. A. Shaginyan. On the limits of indetermination and on the set of limit functions of series in the Walsh system. Sbornik. Mathematics, Tome 24 (1974) no. 2, pp. 257-265. http://geodesic.mathdoc.fr/item/SM_1974_24_2_a3/
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     author = {L. A. Shaginyan},
     title = {On the limits of indetermination and on the set of limit functions of series in the {Walsh} system},
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     year = {1974},
     volume = {24},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_24_2_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The main result of the article is the following Theorem. {\it If a series with respect to the Walsh system is summable $(C,1)$ on a set $E$ of positive measure to a finite function $f(t)$, the subsequence $\{S_{2^n}(t)\}$ of partial sums of this series converges almost everywhere on $E$ to the function $f(t)$.} Bibliography: 8 titles.

[1] S. Kachmazh, G. Shteingauz, Teoriya ortogonalnykh ryadov, Fizmatgiz, Moskva, 1958

[2] A. Zigmund, Trigonometricheskie ryady, t. 1, 2, izd-vo «Mir», Moskva, 1965 | MR

[3] G. Morgenthaler, “Walsh-Fourier series”, Trans. Amer. Math. Soc., 84:2 (1967), 472–507 | DOI | MR

[4] N. J. Fine, “Cesàro summability of Walsh-Fourier series”, Proc. Nat. Acad. Sci. USA, 41:8 (1965), 588–591 | DOI | MR

[5] P. L. Ulyanov, “O ryadakh po sisteme Khaara s monotonnymi koeffitsientami”, Izv. AN SSSR, seriya matem., 28 (1964), 925–950

[6] F. G. Arutyunyan, “O ryadakh po sisteme Khaara”, DAN Arm.SSR, 42:3 (1966), 134–140 | MR | Zbl

[7] V. A. Skvortsov, “O ryadakh Khaara, skhodyaschikhsya po podposledovatelnostyam chastichnykh summ”, DAN SSSR, 183:4 (1968), 784–786 | Zbl

[8] R. F. Gundy, “Martingal theory and pointweise convergence of certain orthogonal series”, Trans. Amer. Math. Soc., 124:2 (1966), 228–248 | DOI | MR | Zbl