Representation of Functions in *L*^{1}_{μ} 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*L*^{1}_{μ} 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.

M. Grigoryan and A. Minasyan, "Representation of Functions in

References

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[2] M. G. Grigorian “On the Representation of Functions by Orthogonal Series in Weighted*L*^{p} 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*L*^{p}_{μ} -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*L*^{p}_{[0.1]},1≤*P*<∞” Proceedings of the American Mathematical Society, Vol. 131, 2002, pp. 1137-1149.

http://dx.doi.org/10.1090/S0002-9939-02-06618-2

[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.

http://dx.doi.org/10.1112/plms/s2-34.1.241

[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. http://dx.doi.org/10.2140/pjm.1955.5.17

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

http://dx.doi.org/10.1090/S0002-9947-1976-0394022-8

[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

[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

[5] M. G. Grigorian and Robert E.Zink,”Subsistems of the Walsh Orthogonal System Whose Multiplicative Completions Are Quasibases for 0

http://dx.doi.org/10.1090/S0002-9939-02-06618-2

[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.

http://dx.doi.org/10.1112/plms/s2-34.1.241

[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. http://dx.doi.org/10.2140/pjm.1955.5.17

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

http://dx.doi.org/10.1090/S0002-9947-1976-0394022-8