For citation:
Sharapudinov I. I. Approximation of Smooth Functions in Lp(x)2π by Vallee–Poussin Means. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 1, pp. 45-49. DOI: 10.18500/1816-9791-2013-13-1-1-45-49, EDN: SMXXIP
Approximation of Smooth Functions in Lp(x)2π by Vallee–Poussin Means
Variable exponent p(x) Lebesgue spaces Lp(x)2π is considered. For f ∈ Lp(x)2π Vallee–Poussin means Vnm(f, x) can be defined as Vnm(f, x) = 1/(m+1)Ʃl=0mSn+l(f, x), where Sk(f, x) –- partial Fourier sum of f(x) of order k. Approximative properties of operators Vnm(f) = Vnm(f, x) are investigated in Lp(x)2π. Let p(x) ≥ 1 be 2π-periodical variable exponent that satisfies Dini–Lipschitz condition. When m = n − 1 and m = n the following estimate is proved: ||f − Vnm(f)||p(·) ≤ (cr(p)/nr )En(f(r))p(·) where En(f(r))p(·) is the best approximation of function f(r)(x) by trigonometric polynomials of order n in Lp(x)2π.
- Шарапудинов И. И. О топологии пространства Lp(x)([0, 1]) // Мат. заметки. 1979. Т. 26, вып. 4. С. 613–632. [Sharapudinov I. I. Topology of the space Lp(x)([0, 1]) // Math. Notes. 1979. Vol. 26, № 4. P. 796—806.]
- Шарапудинов И. И. О базисности системы Хаара в пространстве Lp(x)([0, 1]) и принципе локализации в среднем // Мат. сб. 1986. T. 130(172), № 2(6). С. 275–283. [Sharapudinov I. I. On the basis property of the Haar system in the space Lp(x)([0, 1]) and the principle of localization in the mean // Math. USSR Sb. 1987. Vol. 58, № 1. P. 279–287.]
- Шарапудинов И. И. О равномерной ограниченности в Lp (p = p(x)) некоторых семейств операторов свертки // Мат. заметки.1996. Т. 59, вып. 2. С. 291–302. [Sharapudinov I. I.. Uniform boundedness in Lp (p = p(x)) of some families of convolution operators // Math. Notes. 1996. Vol. 59, № 2. P. 205–212.]
- Шарапудинов И. И. Некоторые вопросы теории приближения в пространствах Lp(x) // Analysis Math. 2007. Vol. 33, № 2. P. 135–153. [Sharapudinov I. I. Some problems of approximation theory in spaces Lp(x) // Analysis Math. 2007. Vol. 33, № 2. P. 135–153.]
- Шарапудинов И. И. О базисности системы полиномов Лежандра в пространстве Lp(x)(−1, 1) переменным показателем p(x)// Мат. сб. 2009. Т. 200, № 1. С. 137–160. [Sharapudinov I. I. The basis property of the Legendre polynomials in the variable exponent Lebesgue space Lp(x)(−1, 1) // Sb. Math. 2009. Vol. 200, № 1. P. 133—156.]
- Guven A., Israfilov D. M. Trigonometric approximation in Generalized Lebesgue spaces Lp(x) // J. of Math. Inequalities. 2010. Vol. 4, № 2. P. 285–299.
- Akgün R. Polynomial approximation of function in weigted Lebesgue and Smirnov spaces with nonstandard growth // Georgian Math.J. 2011. Vol. 18. P. 203–235.
- Akgün R. Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent // Ukrainian Math. J. 2011. Vol. 63, № 1. P. 3–23.
- Akgün R., Kokilashvili V. On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces // Banach J. Math. Anal. 2011. Vol. 5, № 1. P. 70–82.
- Шарапудинов И. И. Некоторые вопросы теории приближения функций тригонометрическими полиномами в Lp(x)2π // Математический форум (Итоги науки. Юг России). 2011. Т. 5. С. 108–117. [Sharapudinov I. I. Some problems in approximation theory by trigonometric polynomials in Lp(x)2π // Math. Forum (Itogi nauki. The South of Russia). 2011. Vol. 5. P. 108–117.]
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