AM  Vol.4 No.11 A , November 2013
Representation of Functions in L1μ Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System
Abstract: Let is the Walsh generalized system. In the paper constructed a weighted space , and series in the Walsh generalized system with monotonically decreasing coefficient such that for each function in the space one can find a subseries that converges to in the weighted and almost everywhere on [0,1].
Cite this paper: M. Grigoryan and A. Minasyan, "Representation of Functions in L1μ Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 6-12. doi: 10.4236/am.2013.411A1002.

[1]   D. E. Men’shov, “Sur la Representation des Fonctions Measurables des Series Trigonometriques,” Sbornik: Mathematics, Vol. 9, 1941, pp. 667-692.

[2]   M. G. Grigorian “On the Representation of Functions by Orthogonal Series in Weighted Lp Spaces,” Studia Mathematica, Vol. 134, No. 3, 1999, pp. 207-216.

[3]   M. Grigoryan, “Modification of Functions, Fourier Coefficients and Nonlinear Approximation,” Sbornik: Mathematics, Vol. 203, No. 3, 2012, pp. 49-78.

[4]   M. G. Grigorian, “On the Lpμ -Strong Property of Orthonormal Systems,” Sbornik: Mathematics, Vol. 194, No. 10 2003, pp. 1503-1532.

[5]   M. G. Grigorian and Robert E.Zink,”Subsistems of the Walsh Orthogonal System Whose Multiplicative Completions Are Quasibases for 0 Lp[0.1],1≤P<∞” Proceedings of the American Mathematical Society, Vol. 131, 2002, pp. 1137-1149.

[6]   V. I. Ivanov, “Representation of Functions by Series in Symmetric Metric Spaces without Linear Functionals,” Trudy MIAN SSSR, Vol. 189, 1989, pp. 34-77.

[7]   V. G. Krotov, “Representation of Measurable Functions by Series with Respect to Faber-Schauder System and Universal Series,” Izvestiya: Mathematics, Vol. 41, No. 1, 1977, pp. 215-229.

[8]   B. I. Golubov, A. F. Efimov and V. A. Skvartsov, “Series and Transformations of Walsh,” Moskow, No. 1987.

[9]   D. E. Men’shov, “On the Partial Sums of Trigonometric Series,” Sbornik: Mathematics, Vol. 20, No. 2, 1947, pp. 197-238.

[10]   R. E. A. C. Paley, “A Remarkable Set of Orthogonal Functions,” Proceedings of the London Mathematical Society, Vol. 34, No. 1, 1932, pp. 241-279.

[11]   A. A. Talalian, “Representation of Measurable Functions by Series,” UMN, Vol. 15, No. 5, 1960, pp. 567-604.

[12]   P. L. Ul’janov, “Representation of Functions by Series and Classes φ(L),” UMN, Vol. 25, No. 2, 1972, pp. 3-52.

[13]   H. E. Chrestenson, “A Class of Generalized Walsh Functions,’’ Pacific Journal of Mathematics, Vol. 45, No. 1, 1955, pp. 17-31.

[14]   W. Young, “Mean Convergence of Generalized WalshFourier Series,” Transactions of the American Mathematical Society, Vol. 218, 1976, pp. 311-320.