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Convergence of Fourier-Walsh Double Series in Weighted \(L_{\mu }^{p}[0,1)^{2}\)

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Analysis and Partial Differential Equations: Perspectives from Developing Countries

Abstract

In this work we discuss the behavior of Fourier coefficients with respect to the Walsh double system, as well as \(L_{\mu }^{p}[0,1)^{2}\)-convergence of the spherical partial sums of the double Fourier-Walsh series after modification of functions.

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References

  1. Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)

    Article  MathSciNet  Google Scholar 

  2. D’yachenko, M.I.: Some problems in the theory of multiple trigonometric series. Uspekhi Mat. Nauk 47(5) (16992), 97–162 (1992) (English transl. In Russian Math. Surveys 47)

    Article  MathSciNet  Google Scholar 

  3. Fefferman, C.: On the divergence of multiple Fourier series. Bull Am. Math. Soc. 77(2), 191–195 (1971)

    Article  MathSciNet  Google Scholar 

  4. Fefferman, C.: The multiple problem for the ball. Ann. Math. 94(2), 330–336 (1971)

    Article  MathSciNet  Google Scholar 

  5. Galoyan, L.N., Grigoryan, M.G., Kobelyan, AKh: Convergence of Fourier series in classical systems. Mat. Sb. 206(7), 55–94 (2015)

    Article  MathSciNet  Google Scholar 

  6. Getsadze, R.D.: On divergence in measure of general multiple orthogonal Furier series. Dokl. Akad. Nauk SSSR 306, 24–25 (1989) (English transl. in Soviet Math. Dokl. 39 1989)

    Google Scholar 

  7. Golubov, B.I., Efimov, A.F., Skvortsov, V.A.: Series and Transformations of Walsh. Moskow (1987) (in Russian)

    Google Scholar 

  8. Golubov, B.I.: Multiple Furier series and integrals. Itogi Nauki i Tekhniki Ser. Mat. Anal. 47(5), 97–162 (1992) (English transl. in J. Soviet Math. 24 1984)

    Google Scholar 

  9. Grigorian, M.G.: On the convergence of Fourier series in the metric of L. Anal. Math. 17, 211–237 (1991)

    Article  MathSciNet  Google Scholar 

  10. Grigorian, M.G.: On the \(L^{p}\)-strong property of orthonormal systems. Math. sb. 194(10), 77–106 (2012) (in Russ., English transl. Sbornik: Math. 194(10), 1503–1532)

    Google Scholar 

  11. Grigorian, M.G., Sargsyan, A.A.: On the coefficients of expansion of elements from C[0,1] space by the Faber-Schauder system. J. Funct. Spaces Appl. 2, 34–42 (2011)

    MathSciNet  Google Scholar 

  12. Grigorian, M.G., Zink, R.E.: Greedy approximation with respect to certain subsystems of the Walsh orthonormal system. Proc. Am. Math. Soc. 134(12), 3495–3505 (2006)

    Article  MathSciNet  Google Scholar 

  13. Grigoryan, M.G.: Modifications of functions, Fourier coefficients and nonlinear approximation. Mat. Sb. 203(3), 49–78 (2012)

    Article  MathSciNet  Google Scholar 

  14. Grigoryan, M.G.: Uniform convergence of the greedy algorithm with respect to the Walsh system. Studia Math. 198(2), 197–206 (2010)

    Article  MathSciNet  Google Scholar 

  15. Grigoryan, M.G., Gogyan, S.L.: On nonlinear approximation with respect to the Haar system and modifications of functions. An. Math. 32, 49–80 (2006)

    Article  Google Scholar 

  16. Grigoryan, M.G., Krotov, V.G.: Luzin’s correction theorem and the coefficients of Fourier expansions in the faber-schauder system. Mat. Zametki 93(2), 172–178 (2013)

    Article  MathSciNet  Google Scholar 

  17. Grigoryan, M.G., Navasardyan, K.A.: On behavior of Fourier coefficients by Walsh systems. J. Contemp. Math. Anal. (Armen. Acad. Sci.) 51(1), 1–13 (2016)

    Article  Google Scholar 

  18. Grigoryan, M.G.: On some properties of orthogonal systems. Izv. Ross. Akad. Nauk, Ser. Mat. 57(5), 75–105 (1993)

    Google Scholar 

  19. Grigoryan, M.G., Sargsyan, A.A.: On the universal functions fr the class Lp[0,1]. J. Funct. Anal. 270, 3111–3133 (2016)

    Article  MathSciNet  Google Scholar 

  20. Grigoryan, M.G., Sargsyan, S.A.: On the Fourier-Vilenkin coefficients. Acta Math. Sci. 37(B)(2), 293–300 (2017)

    Article  MathSciNet  Google Scholar 

  21. Grigoryan, M.G.: Series in the classical systems. LAP LAMBERT Academic Publishing, Saarbruken (2017) (in Russian)

    Google Scholar 

  22. Harris, D.C: Almost everywhere divergence of multiple Walsh-Fourier series. Am. Math. Soc. 101(4) (1987)

    Article  MathSciNet  Google Scholar 

  23. Kolmogorov, A.H.: Sur les fonctions harmoniques conjugees et les series. de Fourier, FM 7, 23–28 (1925)

    Google Scholar 

  24. Luzin, N.N.: On the fundamental theorem of the integral calculus. Mat. Sb. 28, 266–294 (1912). (in Russian)

    Google Scholar 

  25. Men’shov, D.E.: Sur la convergence uniforme des series de Fourier. Mat. Sb. 11(53), 67–96 (1942) (French; Russian)

    Google Scholar 

  26. Men’shov, D.E.: On Fourier series of integrable functions. Trudy Moskov. Mat. Obshch. 1, 5–38 (1952)

    Google Scholar 

  27. Paley, R.E.A.C.: A remarkable set of orthogonal functions. Proc. Lond. Math. Soc. 34, 241–279 (1932)

    Article  Google Scholar 

  28. Riss, M.: Sur les fonctions conjugees. Math. Zeit. 27, 214–244 (1927)

    Google Scholar 

  29. Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45, 5–24 (1923)

    Article  MathSciNet  Google Scholar 

  30. Zhizhiashvili, L.V.: Some problems in the theory of simple and multiple trigonometric and orthogonal series. Uspekhi Mat. Nauk 28(2), 65–119 (1973) (English transl. In Russian, Math. Surveys 28 1973)

    Article  Google Scholar 

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Correspondence to Martin G. Grigoryan .

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Grigoryan, M.G., Grigoryan, T.M., Simonyan, L.S. (2019). Convergence of Fourier-Walsh Double Series in Weighted \(L_{\mu }^{p}[0,1)^{2}\). In: Delgado, J., Ruzhansky, M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics & Statistics, vol 275. Springer, Cham. https://doi.org/10.1007/978-3-030-05657-5_8

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