Geodesic
On the convergence of multiple Haar series
Izvestiya. Mathematics , Tome 78 (2014) no. 1, pp. 90-105.

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We prove that the rectangular and spherical partial sums of the multiple Fourier–Haar series of an individual summable function may behave differently at almost every point, although it is known that they behave in the same way from the point of view of almost-everywhere convergence in the scale of integral classes: $L(\ln^+L)^{n-1}$ is the best class in both cases. We also find optimal additional conditions under which the spherical convergence of a multiple Fourier–Haar series (general Haar series, lacunary series) follows from its convergence by rectangles, and prove that these conditions are indeed optimal.
Keywords: multiple Haar series, spherical convergence, lacunary series.
Mots-clés : convergence by rectangles 
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G. G. Oniani. On the convergence of multiple Haar series. Izvestiya. Mathematics , Tome 78 (2014) no. 1, pp. 90-105. http://geodesic.mathdoc.fr/item/IM2_2014_78_1_a4/

[1] F. Móricz, “On the convergence in a restricted sense of multiple series”, Anal. Math., 5:2 (1979), 135–147 | DOI | MR | Zbl

[2] F. Móricz, “Some remarks on the notion of regular convergence of multiple series”, Acta Math. Hungar., 41:1–2 (1983), 161–168 | DOI | MR | Zbl

[3] O. P. Dzagnidze, “Predstavlenie izmerimykh funktsii dvukh peremennykh dvoinymi ryadami”, Soobsch. AN GSSR, 34 (1964), 277–282 | MR | Zbl

[4] T. Sh. Zerekidze, “Skhodimost kratnykh ryadov Fure–Khaara i silnaya differentsiruemost integralov”, Tr. Tbiliss. matem. in-ta im. A. Razmadze, 76 (1985), 80–99 | MR | Zbl

[5] G. G. Kemkhadze, “O skhodimosti sharovykh chastnykh summ kratnykh ryadov Fure–Khaara”, Tr. Tbiliss. matem. in-ta im. A. Razmadze, 55 (1977), 27–38 | MR | Zbl

[6] G. E. Tkebuchava, “On the divergence of spherical sums of double Fourier–Haar series”, Anal. Math., 20:2 (1994), 147–153 | DOI | MR | Zbl

[7] G. G. Oniani, “On the divergence of multiple Fourier–Haar series”, Dokl. Math., 77:2 (2008), 203–204 | DOI | MR | Zbl

[8] G. G. Oniani, “On the divergence of multiple Fourier–Haar series”, Anal. Math., 38:3 (2012), 227–247 | DOI | MR | Zbl

[9] G. G. Oniani, “On the relationship between rectangular convergence and spherical convergence of multiple Haar series”, Russian Math. Surveys, 67:1 (2012), 186–187 | DOI | DOI | MR | Zbl

[10] G. G. Oniani, “On the regular convergence by rectangles of multiple Fourier–Haar series”, Bull. Georgian Natl. Acad. Sci. (N.S.), 6:2 (2012), 31–33 | MR

[11] A. Zygmund, Trigonometric series, v. I, Cambridge Univ. Press, New York, 1959 | MR | MR | Zbl | Zbl

[12] G. Alexits, Convergence problems of orthogonal series, Pergamon Press, New York–Oxford–Paris, 1961 | MR | MR | Zbl | Zbl

[13] B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989 | MR | MR | Zbl | Zbl

[14] M. de Guzman, Differentiation of integrals in $\mathbb{R}^n$, Lecture Notes in Math., 481, Springer-Verlag, Berlin–New York, 1975 | MR | MR | Zbl

[15] S. Saks, Theory of the integral, Dover Publ., New York, 1937 | MR | Zbl