RU  |  EN

ISSN  1684-792X

Issues:


Sections:

Issue 47

MATHEMATICS AND PHYSICS

UDK: 517.9

CONTEMPORARY PROBLEMS OF THE THEORY OF APPROXIMACY AND SPACE L(E, p(x), mu)

SHARAPUDINOV Idris Idrisovich


Some questions concerning approximate calculation of integrals are researched in the present paper. The problem of error estimations of quadrature formulas for the functions from classes , consist ing of functions , given on , having absolutely continuous derivative and derivative of order , having property is considered. Such classes were introduced in the author’s works in connection with the problem of error estimations of quadrature formulas. Some new classes of functions, having essentially variable behavior of smoothness on the given interval , and the obtained exact error estimations of the so called complicated quadrature formulas for these new classes are also considered. When solving this problem spaces with variable exponent and dual spaces , where , appear naturally. Precisely, it expresses, in the terms of these spaces, the exact values of the estimations of the quadrature formulas taking into account essentially variable behavior of smoothness of functions liable to approximate integration on interval . It allows to get more exact error estimations of complicated quadrature formulas using the next values where is a linear capsule of space .

Keywords: пространства Лебега с переменным показателем; пространства Соболева с переменным показатем; квадратурные формулы; приближение функций; variable exponent Lebesgue spaces; variable exponent Sobolev spaces; quadrature formulas; function approximation.

Pages: 5 - 21

Date: 15.03.2012

Bibliography:

  • Orlicz W. Uber konjugierte Exponentenfolgen // Studia Math. 1931. Vol. 3. P. 200–212.
  • Nakano H. Modulared Semi-ordered Linear Spaces. Tokyo: Maruzen Co., Ltd., 1950.
  • Nakano H. Topology and Topological Linear Spaces. Tokyo: Maruzen Co., Ltd., 1951.
  • Musielak J. Orlicz Spaces and Modular Spaces. Berlin: Springer-Verlag, 1983.
  • Musielak J. and Orlicz W. On modular spaces // Studia Math. 1959. Vol. 18. P. 49–65.
  • Tsenov I. V. Generalization of the problem of best approximation of a function in the space // (Russian) Uch. Zap. Dagestan Gos. Univ. 1961. Vol. 7. P. 25–37.
  • Колмогоров А. Н. Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes // Studia Math. 1934. Vol. 5. P. 29–33.
  • Шарапудинов И.И. О топологии пространства // Матем. заметки. 1979. Т. 26. Вып. 4. С. 613–632.
  • Шарапудинов И.И. О базисности системы Хаара в пространстве и принципе локализации в среднем // Матем. сборник. 1986. Т. 130 (172). № 2 (6). С. 275–283.
  • Шарапудинов И.И. О равномерной ограниченности в некоторых семейств операторов свертки // Матем. заметки. 1996. Т. 59. Вып. 2. С. 291–302.
  • Diening L. Maximal function on generalized Lebesgue spaces Lp(·) // Math. Inequal. Appl. 2004. Vol. 7. P. 245–253.
  • Diening L. and Ruħička M. Calderon-Zygmund operators on generalized Lebesgue spaces and problems related to fluid dynamics // J. Reine Angew. Math. 2003. N 563. P. 197–220.
  • Diening L., Hästö P. and Nekvinda A. Open problems in variable exponent Lebesgue and Sobolev spaces // Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28 – June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha.
  • Lebesgue and Sobolev spaces with variable exponent. Lecture Notes in Mathematics 2017 / L. Diening, Р. Harjulehto, Р. Hasto, М. Ruħička // Springer-Verlag Berlin and Heidelberg, 2011.
  • Zhikov V.V. Averaging of functionals of the calculus of variations and elasticity theory // Math. USSR Izv. 1987. Vol. 29 N 1. P. 33–66. [Translation of Izv. Akad. Nauk SSSR Ser. Mat. 1986. Vol. 50. N 4. P. 675–710, 877.]
  • Zhikov V.V. Meyer-type estimates for solving the nonlinear Stokes system // Differ. Equ. 1997. Vol. 33. N 1. P. 108–115. [Translation of Differ. Uravn. 1997 Vol. 33. N 1. P. 107–114, 143].
  • Zhikov V.V. On some variational problems // Russian J. Math. Phys. 1997. Vol. 5. N 1. P. 105–116 (1998).
  • Kokilashvili V., Samko N. and Samko S. Singular operators in variable spaces with oscillating weights // Math. Nachr. 2007. Vol. 280. P. 1145–1156.
  • Kokilashvili V. and Samko S. Singular integral equations in the Lebesgue spaces with variable exponent // Proc. A. Razmadze Math. Inst. 2003. Vol. 131. P. 61–78.
  • Kokilashvili V. and Samko S. Singular integrals in weighted Lebesgue spaces with variable exponent // Georgian Math. J. 2003. Vol. 10. P. 145–156.
  • Kokilashvili V. and Samko S. Weighted boundedness in Lebesgue spaces with variable exponents of classical operators on Carleson curves // Proc. A. Razmadze Math. Inst. 2005. Vol. 138. P. 106–110.
  • Kokilashvili V. and Samko S. Singular Integrals in Weigted Lebesgue Spaces with Variable Exponent // Georgian Math. J. 2003. Vol. 10. N 1. P. 145–156.
  • Samko S. Convolution type operators in Lp(x) // Integr. Transform. Spec. Funct. 1998. Vol. 7. N 1, 2. P. 123–144.
  • Samko S. Hardy inequality in the generalized Lebesgue spaces // Fract. Calc. Appl. Anal. 2003. N 6. P. 355–362.
  • Шарапудинов И.И. Некоторые вопросы теории приближения в пространствах // Analysis Mathematica. 2007. T. 33. № 2. С. 135–153.
  • Шарапудинов И.И. О базисности системы полиномов Лежандра в пространстве с переменным показателем // Матем. сб. 2009. Т. 200. № 1. С. 137–160.
  • Шарапудинов И.И. Приближение функций в метрике пространства и квадратурные формулы // Constructive function theory'81. Procedings of the International Conference on Constructive Function Theory. Varna, June 1–5, 1981. С. 189–193.
  • Guven A. and Israfilov D. M. Trigonometric approximation in Generalized Lebesgue spaces // Journal of Math. Inequalities. 2010. Vol. 4. N 2. P. 285–299.
  • Akgun R. Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth // Georgian Math. J. 2011. N 18. P. 203–235. DOI 10.1515/GMJ.2011.0022
  • Akgun R. Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent // Ukr. Math. Journal. 2011. Vol. 63. N 1. (Ukrainian Original. 2011. Vol. 63. N 1).
  • Akgun R. and Kokilashvili V. On converse theorems of trigonometric approximation in weghted variable exponent Lebesgue spaces // Banach J. Math. Anal. 2011. Vol. 5 N 1. P. 70–82.
  • Guven A. Trigonometric approximation by matrix transforms in Lp(x) // Anal. and Appl. 2012. Vol. 10. N 1. Р. 47–65.
  • Akgun R. and Kokilashvili V. The refined direct and converse inequalities of trigonometric approximation in weighted variable exponent Lebesgue spaces // Georgian Math. J. 2011. Vol. 18. N 3. P. 399–423.
  • Шарапудинов И.И. Некоторые вопросы теории приближения функций тригонометрическими полиномами в // Математический форум. Т. 5. Исследования по математическому анализу и дифференциальным уравнениям. Владикавказ: ЮМИ ВНЦ РАН и РСО-А. 2011. С. 108–116.
  • Никольский С.М. Квадратурные формулы. М.: Наука, 1988.

Download full text


Back to Table of Contents