Research ArticleOpen AccessGoogle Scholar indexed
Representation of Functions in <i>L</i><sup>1</sup><sub style="margin-left:-6px"><i>μ</i></sub> Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System
Department of Physics, Yerevan State University, Yerevan, Armenia
Department of Physics, Yerevan State University, Yerevan, Armenia
- 1 Department of Physics, Yerevan State University, Yerevan, Armenia
- 2 Department of Physics, Yerevan State University, Yerevan, Armenia
Applied Mathematics·Volume 04 (2013)·Pages 6–12·Published 16 October 2013·DOI10.4236/am.2013.411A1002
Copy link · social · email
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].
KeywordsOrthonormal SystemConvergenceFunctional Series
- D. E. Men’shov, “Sur la Representation des Fonctions Measurables des Series Trigonometriques,” Sbornik: Mathematics, Vol. 9, 1941, pp. 667-692.
- 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.
- M. Grigoryan, “Modification of Functions, Fourier Coefficients and Nonlinear Approximation,” Sbornik: Mathematics, Vol. 203, No. 3, 2012, pp. 49-78.
- M. G. Grigorian, “On the L p μ -Strong Property of Orthonormal Systems,” Sbornik: Mathematics, Vol. 194, No. 10 2003, pp. 1503-1532.
- 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 http://dx.doi.org/10.1090/S0002-9939-02-06618-2
- V. I. Ivanov, “Representation of Functions by Series in Symmetric Metric Spaces without Linear Functionals,” Trudy MIAN SSSR, Vol. 189, 1989, pp. 34-77.
- 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.
- B. I. Golubov, A. F. Efimov and V. A. Skvartsov, “Series and Transformations of Walsh,” Moskow, No. 1987.
- D. E. Men’shov, “On the Partial Sums of Trigonometric Series,” Sbornik: Mathematics, Vol. 20, No. 2, 1947, pp. 197-238.
- 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
- A. A. Talalian, “Representation of Measurable Functions by Series,” UMN, Vol. 15, No. 5, 1960, pp. 567-604.
- P. L. Ul’janov, “Representation of Functions by Series and Classes φ(L),” UMN, Vol. 25, No. 2, 1972, pp. 3-52.
- 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
- 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