vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2021-57-3-319-329vestifm-600Research ArticleМАТЕМАТИКАMATHEMATICSОб одном обобщении квадратурной формулы ЭрмитаOn one generalization of the Hermite quadrature formulahttps://orcid.org/0000-0002-1265-1965РовбаЕ. А.RoubaY. A.

Ровба Евгений Алексеевич – доктор физико-математических наук, профессор, заведующий кафедрой фундаментальной и прикладной математики

ул. Ожешко, 22, 230023, г. Гродно

Yauheni A. Rouba – Dr. Sc. (Physics and Mathematics), Professor, Head of the Department of Fundamental and Applied Mathematics

22, Ozheshko Str., 230023,  Grodno

rovba.ea@gmail.comhttps://orcid.org/0000-0001-9054-8691СмотрицкийК. А.SmatrytskiK. A.

Смотрицкий Константин Анатольевич – кандидат физико-математических наук, доцент, доцент кафедры фундаментальной и прикладной математики

ул. Ожешко, 22, 230023, г. Гродно

Kanstantin A. Smatrytski – Ph. D. (Physics and Mathematics), Associate Professor of the Department of Fundamental and Applied Mathematics

22, Ozheshko Str., 230023, Grodno

k_smotritski@mail.ruhttps://orcid.org/0000-0002-4312-0462ДирвукЕ. В.DirvukY. V.

Дирвук Евгений Владимирович – кандидат физико-математических наук, доцент кафедры системного программирования и компьютерной безопасности

ул. Ожешко, 22, 230023, г. Гродно

Yauheni V. Dirvuk – Ph. D. (Physics and Mathematics), Associate Professor of the Department of System Programming and Computer Security

22, Ozheshko Str., 230023, Grodno

dirvuk@gmail.comГродненский государственный университет им. Я. КупалыYanka Kupala State University of Grodno965Гродненский государственный университет им. Я. КупалыYanka Kupala State University of Grodno202107102021573319329Copyright © Ровба Е.А., Смотрицкий К.А., Дирвук Е.В., 20212021Ровба Е.А., Смотрицкий К.А., Дирвук Е.В.Rouba Y.A., Smatrytski K.A., Dirvuk Y.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/600

Целью данной работы является изучение нового подхода к построению квадратурных формул интерполяционно-рационального типа на отрезке. Проведен краткий анализ результатов по теме исследования, где основное внимание уделено работам математиков белорусской школы по теории аппроксимации – квадратурным формулам Гаусса, Лобатто, Радо с узлами в нулях рациональных дробей Чебышева – Маркова. Определяются рациональные дроби на отрезке, обобщающие классические ортогональные многочлены Якоби с одним весом, и описываются некоторые их свойства. Один из основных результатов работы состоит в построении квадратурных формул с узлами в нулях введенных рациональных дробей, вычислении их коэффициентов в явном виде, оценке остатка. Ему предшествуют некоторые вспомогательные утверждения, описывающие свойства специальных рациональных функций. Для доказательства используются классические методы математического анализа, теории приближений и теории функций комплексного переменного. Проводится численный анализ эффективности построенных квадратурных формул. При этом выбор параметров, от которых зависят узлы квадратурных формул, производится несколькими стандартными способами. Полученные результаты могут быть применены для дальнейшего исследования рациональных квадратурных формул, а также в численном анализе.

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.

приближениеинтерполяциярациональные дробиквадратурные формулычисленный анализapproximationinterpolationrational fractionsquadrature formulasnumerical analysisReferencesDeckers K., Mougaida A., Belhadjsalah H. Algorithm 973: extended rational Fejér quadrature rules based on Chebyshev orthogonal rational functions. ACM Transactions on Mathematical Software, 2017, vol. 43, no. 4, pp. 15–66. https://doi.org/10.1145/3054077Deckers K., Mougaida A., Belhadjsalah H. Algorithm 973: extended rational Fejér quadrature rules based on Chebyshev orthogonal rational functions. ACM Transactions on Mathematical Software, 2017, vol. 43, no. 4, pp. 15–66. https://doi.org/10.1145/3054077Deckers K. Christoffel–Darboux-type formulae for orthonormal rational functions with arbitrary complex poles. IMA Journal of Numerical Analysis, 2015, vol. 35, no. 4, pp. 1842–1863. https://doi.org/10.1093/imanum/dru049Deckers K. Christoffel–Darboux-type formulae for orthonormal rational functions with arbitrary complex poles. IMA Journal of Numerical Analysis, 2015, vol. 35, no. 4, pp. 1842–1863. https://doi.org/10.1093/imanum/dru049Deckers K., Bultheel A., Perdomo-Pío F. Rational Gauss-Radau and Szegö-Lobatto quadrature on the interval and the unit circle respectively. Jaen Journal on Approximation, 2011, vol. 3, no. 1, pp. 15–66.Deckers K., Bultheel A., Perdomo-Pío F. Rational Gauss-Radau and Szegö-Lobatto quadrature on the interval and the unit circle respectively. Jaen Journal on Approximation, 2011, vol. 3, no. 1, pp. 15–66.Rouba Y. A. Quadrature formulas of interpolation rational type. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 1996, vol. 40, no. 3, pp. 42–46 (in Russian).Rouba Y. A. Quadrature formulas of interpolation rational type. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 1996, vol. 40, no. 3, pp. 42–46 (in Russian).Rouba Y. A. On one orthogonal system of rational functions and quadratures of Gauss type. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 1998, no. 3, pp. 31–35 (in Russian).Rouba Y. A. On one orthogonal system of rational functions and quadratures of Gauss type. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 1998, no. 3, pp. 31–35 (in Russian).Rouba Y. A., Dirvuk Y. V. On one quadrature formula of interpolation rational type with respect to Chebyshev – Markov nodes. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2014, vol. 58, no. 5, pp. 23–29 (in Russian).Rouba Y. A., Dirvuk Y. V. On one quadrature formula of interpolation rational type with respect to Chebyshev – Markov nodes. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2014, vol. 58, no. 5, pp. 23–29 (in Russian).Dirvuk Y. V., Smatrytski K. A. Rational quadrature formulas of Radau type. Vestnik BGU. Seriya 1, Fizika. Matematika. Informatika = Proceedings of BSU. Series 1. Physics. Mathematics. Informatics, 2014, no. 1, pp. 87–91 (in Russian).Dirvuk Y. V., Smatrytski K. A. Rational quadrature formulas of Radau type. Vestnik BGU. Seriya 1, Fizika. Matematika. Informatika = Proceedings of BSU. Series 1. Physics. Mathematics. Informatics, 2014, no. 1, pp. 87–91 (in Russian).Rusak V. N., Rybachenka I. V. Chebyshev – Markov’s cosine-fractions in the approximate integration. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2014, no. 3, pp. 15–20 (in Russian).Rusak V. N., Rybachenka I. V. Chebyshev – Markov’s cosine-fractions in the approximate integration. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2014, no. 3, pp. 15–20 (in Russian).Rusak V. N., Grib N. V. Chebyshev – Markov sine-fractions in approximate integration. Vestsі BDPU. Seryja 3. Fіzіka. Matematyka. Іnfarmatyka. Bіyalogіya. Geagrafіya. = Proceedings of BSPU. Series 3. Physics. Mathematics. Informatics. Biology. Geography, 2015, no. 2, pp. 17–20 (in Russian).Rusak V. N., Grib N. V. Chebyshev – Markov sine-fractions in approximate integration. Vestsі BDPU. Seryja 3. Fіzіka. Matematyka. Іnfarmatyka. Bіyalogіya. Geagrafіya. = Proceedings of BSPU. Series 3. Physics. Mathematics. Informatics. Biology. Geography, 2015, no. 2, pp. 17–20 (in Russian).Markov A. A. Selected Works on the Theory of Continuous Fractions and the Theory of Functions with the Least Deviation from Zero. Moscow, Leningrad, GITTL Publ., 1948. 413 p. (in Russian).Markov A. A. Selected Works on the Theory of Continuous Fractions and the Theory of Functions with the Least Deviation from Zero. Moscow, Leningrad, GITTL Publ., 1948. 413 p. (in Russian).Rusak V. N. Rational Functions as Approximation Apparatus. Minsk, BSU Publ., 1979. 176 p. (in Russian).Rusak V. N. Rational Functions as Approximation Apparatus. Minsk, BSU Publ., 1979. 176 p. (in Russian).Szabados J., Vertesi P. Interpolation of Functions. World Scientific, 1990. 305 p.Szabados J., Vertesi P. Interpolation of Functions. World Scientific, 1990. 305 p.Rouba Y. A., Smatrytski K. A. Convergence in the mean of rational interpolating processes in the zeroes of Bernstein fractures. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2010, no. 3, pp. 5–9 (in Russian).Rouba Y. A., Smatrytski K. A. Convergence in the mean of rational interpolating processes in the zeroes of Bernstein fractures. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2010, no. 3, pp. 5–9 (in Russian).Natanson I. P. Constructive Theory of Functions. Moscow, Leningrad, Gostechizdat Publ., 1949. 688 p. (in Russian).Natanson I. P. Constructive Theory of Functions. Moscow, Leningrad, Gostechizdat Publ., 1949. 688 p. (in Russian).

The authors declare that there are no conflicts of interest present.