On one generalization of the Hermite quadrature formula
https://doi.org/10.29235/1561-2430-2021-57-3-319-329
Abstract
In this paper we propose a new approach to the construction of quadrature formulas of interpolation rational type on an interval. In the introduction, a brief analysis of the results on the topic of the research is carried out. Most attention is paid to the works of mathematicians of the Belarusian school on approximation theory – Gauss, Lobatto, and Radau quadrature formulas with nodes at the zeros of the rational Chebyshev – Markov fractions. Rational fractions on the segment, generalizing the classical orthogonal Jacobi polynomials with one weight, are defined, and some of their properties are described. One of the main results of this paper consists in constructing quadrature formulas with nodes at zeros of the introduced rational fractions, calculating their coefficients in an explicit form, and estimating the remainder. This result is preceded by some auxiliary statements describing the properties of special rational functions. Classical methods of mathematical analysis, approximation theory, and the theory of functions of a complex variable are used for proof. In the conclusion a numerical analysis of the efficiency of the constructed quadrature formulas is carried out. Meanwhile, the choice of the parameters on which the nodes of the quadrature formulas depend is made in several standard ways. The obtained results can be applied for further research of rational quadrature formulas, as well as in numerical analysis.
About the Authors
Y. A. Rouba
Yanka Kupala State University of Grodno965
Belarus
Belarus
Yauheni A. Rouba – Dr. Sc. (Physics and Mathematics), Professor, Head of the Department of Fundamental and Applied Mathematics
22, Ozheshko Str., 230023, Grodno
K. A. Smatrytski
Yanka Kupala State University of Grodno
Belarus
Belarus
Kanstantin A. Smatrytski – Ph. D. (Physics and Mathematics), Associate Professor of the Department of Fundamental and Applied Mathematics
22, Ozheshko Str., 230023, Grodno
Y. V. Dirvuk
Yanka Kupala State University of Grodno
Belarus
Belarus
Yauheni V. Dirvuk – Ph. D. (Physics and Mathematics), Associate Professor of the Department of System Programming and Computer Security
22, Ozheshko Str., 230023, Grodno
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