Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes
https://doi.org/10.29235/1561-2430-2019-55-4-391-405
Abstract
In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:
a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;
b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;
c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.
The results can be applied in theoretical research and numerical methods.
About the Authors
E. A. Rovba
Yanka Kupala State University of Grodno
Belarus
Belarus
Evgeniy A. Rovba – Dr. Sc. (Physics and Mathematics), Professor, Head of the Department of Fundamental and Applied Mathematics, Faculty of Mathematics and Informatics.
22, Ozheshko Str., 230023, Grodna
V. Yu. Medvedeva
Yanka Kupala State University of Grodno
Belarus
Belarus
Victoria Yu. Medvedeva – Undergraduate.
22, Ozheshko Str., 230023, Grodna
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