Abstract
It is proved that for any dimension n ≥ 2, L(ln+ L)n−1 is the widest integral class in which the almost everywhere convergence of spherical partial sums of multiple Fourier-Haar series is provided. Moreover,it is shown that the divergence effects of rectangular and spherical general terms of multiple Fourier-Haar series can be achieved simultaneously on a set of full measure by an appropriate rearrangement of values of arbitrary summable function f not belonging to L(ln+ L)n−1.
Реэюме
Доказано, что для любой размерности n ≥ 2, L(ln+ L)n−1 является наиболее широким интегральным классом, в котором обеспечено почти всюду сходимость кратных рядов Фурье-Хаара, Более того, показано, что можно одновременно добиться Эффектов расходимости как прямоугольных, так и сферических общих членов кратных рядов Фурье-Хаара на множестве полной меры путем подходящей перестановки значений произвольной суммируемой функции f, не принадлежащей классу L(ln+ L)n−1.
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References
G. Alexits, Convergence problems of orthogonal series, Pergamon Press (New York- Oxford-Paris, 1961).
R. D. Getsadze, On divergence of the general terms of the double Fourier-Haar series, Arch. Math., 86(2006), 331–339.
R. D. Getsadze, Divergence of spherical general terms of double Fourier series, J. Fourier Anal. Appl., 12(2006), 597–604.
M. de Guzmán, it Differentiation of integrals in ℝn, Lecture Notes in Mathematics 481, Springer (Berlin, 1975).
M. de Guzmán, An inequality for the Hardy-Littlewood maximal operator with respect to a product of differentiation bases, Studia Math., 49(1974), 185–194.
B. Jessen, J. Marcinkiewicz, and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math., 25(1935), 217–234.
B. S. Kashin and A. A. Saakyan, Orthogonal series, Nauka (Moscow, 1984) (in Russian).
G. G. Kemkhadze, On the convergence of the spherical partial sums of multiple Fourier-Haar series, Trudy Tbiliss. Mat. Inst. Razmadze, 55(1977), 27–38 (in Russian).
G. G. Kemkhadze, On the divergence of spherical partial sums of double Fourier-Haar series, Trudy Gruz. Politech. Inst., Mat. Analiz, 3(1985), 42–48 (in Russian).
G. G. Oniani, On the Fourier-Haar series convergence with respect to sets which are homothetic to the given set, Proc. A. Razmadze Math. Inst., 126(2001), 126–128.
G. G. Oniani, On divergence of multiple Fourier-Haar series, Dokl. Akad. Nauk, 419(2008), no. 2, 169–170 (in Russian); English translation: Dokl. Math., 77(2008), no. 2, 203–204.
G. G. Oniani, Differentiation of Lebesgue integrals, Tbilisi University Press (Tbilisi, 1998) (in Russian).
S. Saks, On the strong derivatives of functions of intervals, Fund. Math., 25(1935), 235–252.
A. M. Stokolos, Differentiation of integrals of equimeasurable functions, Mat. Zametki, 37(1985), no. 5, 667–675 (in Russian); English translation Math. Notes, 37(1985), 364–368.
G. E. Tkebuchava, On the divergence of the spherical sums of double Fourier-Haar series, Analysis Math., 20(1994), 147–153.
T. S. Zerekidze, The convergence of multiple Fourier-Haar series and the strong differentiability of integrals, Trudy Tbiliss. Mat. Inst. Razmadze, 76(1985), 80–99 (in Russian).
T. S. Zerekidze, The solution of one problem in the theory of differentiation of integrals, Proc. A. Razmadze Math. Inst., 133(2003), 131–155.
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Oniani, G.G. On the divergence of multiple Fourier-Haar series. Anal Math 38, 227–247 (2012). https://doi.org/10.1007/s10476-012-0305-2
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DOI: https://doi.org/10.1007/s10476-012-0305-2