About the absolute value function with different nodes of Lagrange interpolation
Abstract
Lagrange interpolation processes are considered for the following matrixes of interpolation nodes: the matrix of Chebyshev polynomial roots of the 1st kind, the matrix of Legendre polynomials roots, and the extended matrix of Legendre polynomials roots. For these matrixes the uniform convergence of Lagrange process of interpolation for the absolute value function proved. Also, we receive estimates on the order of convergence for each of these matrixes. To ensure the quality of convergence, the endpoints of the segment were added as nodes to the matrix of Legendre roots. However, for the absolute value function the order of convergence of the Legendre process does not change, but improves by approximately 8 times. For comparison, the negative result of equidistant nodes is taken.
References
[1] Bernstein S. N., “Certain remarks about interpolation [in French],” Charikov, Comm. Soc. Math., 2, 49–61 (1916).
[2] Natanson G. I., The Constructive Theory of Functions [in Russian], Gostekhizdat, Moscow (1949).
[3] Runck P. O., “Uber Konvergenzfragen bei Polynominterpolation mit ¨aqui ¨ distanten Knoten. I–II,” J. Reine Angew. Math., 208, 51–69 (1961); 210, 175–204 (1962).
[4] Rabkin E. L. and Shapiro E. P., “A divergent process of interpolation [in Russian],” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 103–110 (1971).
[5] Natanson G. I., “Estimate of lower and upper bounds for Lebesgue function of Lagrange interpolation processes with Jacobi nodes [in Russian],” Izv. Vyssh. Uchebn. Zaved., Mat., No. 11, 67–74 (1967).
[6] Agahanov S. A., “Assessment of Lebesgue function for interpolation process in roots of Jacobi polynomials [in Russian],” Izv. Vyssh. Uchebn. Zaved., Mat., No. 11, 3–6 (1967).
[7] Privalov A. A., “About uniform convergence of Lagrange interpolation processes,” Math. Notes, 39, No. 2, 228–243 (1986).
[8] Privalov A. A., “Criterion of uniform convergence of Lagrange interpolation processes [in Russian],” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 49–59 (1986).
[9] Khokholov V. B., “Comparison of tests of convergence of Lagrange interpolation processes in Jacobi matrixes on [−1; 1] [in Russian],” Dep. VINITI 29.12.85, No. 8972, 40 p.
[10] Khokholov V. B., “Uniform convergence of Lagrange interpolation processes [in Russian],” in: Dynamics of Control Systems and Evaluation Processes, Mezhvuz. Sb. Nauch. Trudov, Yakutsk, 1991, pp. 52–64.
[11] Turetsky A. H., Interpolation Theory in Problems [in Russian], Vysheyshaya Shkola, Minsk (1968).
[12] Suetin P. K., Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1979).
[13] Daugavet I. K., Introduction to the Theory of Approximation of Functions [in Russian], Izdat. Leningr. Univ., Leningrad (1977).
[14] Szego G., Orthogonal Polynomials, Amer. Math. Soc., New York (1959).
[15] Khokholov V. B. and Kirillina A. A., “About one constant of Lebesgue [in Russian],” in: Tez. Dokl. VII Resp. Nauch.-Prakt. Konf. Molodykh Uchyonykh i Spetsialistov, Yakutsk, 1988, pp. 47.
This work is licensed under a Creative Commons Attribution 4.0 International License.