Abstract
The paper deals with the question of convergence of multiple Fourier-Haar series with partial sums taken over homothetic copies of a given convex bounded set \(W\subset\mathbb{R}_+^n\) containing the intersection of some neighborhood of the origin with \(\mathbb{R}_+^n\). It is proved that for this type sets W with symmetric structure it is guaranteed almost everywhere convergence of Fourier-Haar series of any function from the class L(ln+L)n−1.
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References
B. S. Kashin and A. A. Saakyan, Orthogonal Series (Amer. Math. Soc., Providence, 1989).
G. H. Hardy, “On the convergence of certain multiple series”, Proc. Cambridge Philos. Soc., 19, 86–95, 1916–1919.
F. Moricz, “On the convergence in a restricted sense of multiple series”, Anal. Math., 5(2), 135–147, 1979.
M. de Guzmán, Differentiation of integrals in ℝn, Lecture Notes in Math., vol. 481 (Springer, Berlin-New York, 1975).
T. Sh. Zerekidze, “Convergence of multiple Fourier-Haar series and strong differentiablity of integrals”, Trudy Tbilis. Mat. Inst. Razmadze, 76, 80–99, 1985.
G. A. Karagulyan, “Divergence of double Fourier series in complete orthonormal systems”, Soviet J. Contemporary Math. Anal., 24(2), 44–56, 1989.
G. Gat and G. Karagulyan, “On Convergence Properties of Tensor Products of Some Operator Sequences”, The Journal of Geometric Analysis, 26(4), 3066–3089, 2016.
G. G. Kemkhadze, “Convergence of spherical partial sums of multiple Fourier-Haar series”, Trudy Tbilis. Mat. Inst. Razmadze Akad. Nauk Gruz. SSR, 55, 27–38, 1977.
G. E. Tkebuchava, “On the divergence of spherical sums of double Fourier-Haar series”, Anal. Math. 20(2), 147–153, 1994.
G. G. Oniani, “On the divergence of multiple Fourier-Haar series, Anal. Math., 38(3), 227–247, 2012.
G. G. Oniani, “On the convergence of multiple Haar series”, Izv. Math., 78 (1)78 (1), 90–105, 2014.
G. G. Oniani, “The convergence of double Fourier-Haar series over homothethic copies of sets”, Sb. Mat., 205(7), 983–1003, 2014.
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Russian Text © The Author(s), 2019, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2019, No. 5, pp. 70–81.
The research is supported by Shota Rustaveli National Science Foundation (project no. 217282).
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Oniani, G.G., Tulone, F. On the Almost Everywhere Convergence of Multiple Fourier-Haar Series. J. Contemp. Mathemat. Anal. 54, 288–295 (2019). https://doi.org/10.3103/S1068362319050054
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DOI: https://doi.org/10.3103/S1068362319050054