ABOUT ONE SYSTEM OF THE CHEBYSHEV–MARKOV RATIONAL FRACTIONS
Abstract
The Chebyshev–Markov rational fractions have a number of remarkable properties and are one of the main and, certainly, important elements in the theory of approximation of functions. The Chebyshev–Markov fractions are the integrant apparatus for creation of interpolation rational functions and quadrature formulas. However in the context of the orthogonal Fourier series they are not used as generally they have no property of orthogonality.
The present article considers the system of the Chebyshev–Markov rational fractions with a special choice of the parameters for its definition. In the first part of the present article, the elements of the system are created, some of their representations are specified, and it is proved that there is a weight, at which the studied system is orthogonal on a piece [–1, 1]. In the second part of the work, the Dirichlet integral is created. In the third part of the present article, the coefficients of the Fourier series expansion of the function |x| in the considered system are found in explicit form. In the fourth part, the estimate of the function |x| by means of the partial sums of its Fourier series is investigated. In particular, its accuracy is proved. In the closing part, the asymptotic estimate of the approximation by the partial sums on a piece is obtained as a whole and when the approximation is carried out outside a singular point. Precise constants of these estimates are found.
The present article considers the system of the Chebyshev–Markov rational fractions with a special choice of the parameters for its definition. In the first part of the present article, the elements of the system are created, some of their representations are specified, and it is proved that there is a weight, at which the studied system is orthogonal on a piece [–1, 1]. In the second part of the work, the Dirichlet integral is created. In the third part of the present article, the coefficients of the Fourier series expansion of the function |x| in the considered system are found in explicit form. In the fourth part, the estimate of the function |x| by means of the partial sums of its Fourier series is investigated. In particular, its accuracy is proved. In the closing part, the asymptotic estimate of the approximation by the partial sums on a piece is obtained as a whole and when the approximation is carried out outside a singular point. Precise constants of these estimates are found.
About the Authors
Y. A. Rovba
Yanka Kupala State University of Grodno
Belarus
D. Sc. (Physics and Mathematics), Professor, Head of the Department
Belarus
D. Sc. (Physics and Mathematics), Professor, Head of the Department
P. G. Potsejko
Yanka Kupala State University of Grodno
Belarus
Postgraduate student, Senior lecturer
Belarus
Postgraduate student, Senior lecturer
References
1. Markov A. A. Selected works on the theory of continued fractions and the theory of functions least deviating from zero. Moscow, Leningrad, 1948. 411 p.
2. Gradshtejn I. S., Ryzhyk I. M. Tables of integrals, sums, series and products. Moscow, 1963. 1100 p. (in Russian)
3. Natanson I. P. Constructive theory of functions. Moscow, Leningrad, 1949. 684 p. (in Russian)
4. Evgrafov M. A. Asymptotic estimates and entire functions. Moscow, Nauka Publ., 1979. 320 p. (in Russian)
5. Fedorjuk M. V. Asymptotics. Integrals and Series. Moscow, Nauka Publ., 1987. 544 p. (in Russian)
6. Zorich V. A. Mathematical analysis. Part II. Moscow, Nauka Publ., 1984. 640 p. (in Russian)
7. Rovba E. A., Mikulich E. G. Constants in the approximation of |x| using the rational interpolation processes. Doklady Natsional’noi akademii nauk Belarusi [Doklady of the National Academy of Sciences of Belarus], 2009, vol. 53, no. 6, pp. 11–15 (in Russian)
Review
Views: 813
Email this article 