Extremal polynomials related to Zolotarev polynomials
- Authors: Agafonova I.V.1, Malozemov V.N.1
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Affiliations:
- St. Petersburg State University
- Issue: Vol 93, No 2 (2016)
- Pages: 164-165
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/223480
- DOI: https://doi.org/10.1134/S1064562416020113
- ID: 223480
Cite item
Abstract
Algebraic polynomials bounded in absolute value by M > 0 in the interval [–1, 1] and taking a fixed value A at a > 1 are considered. The extremal problem of finding such a polynomial taking a maximum possible value at a given point b < −1 is solved. The existence and uniqueness of an extremal polynomial and its independence of the point b < −1 are proved. A characteristic property of the extremal polynomial is determined, which is the presence of an n-point alternance formed by means of active constraints. The dependence of the alternance pattern, the objective function, and the leading coefficient on the parameter A is investigated. A correspondence between the extremal polynomials in the problem under consideration and the Zolotarev polynomials is established.
About the authors
I. V. Agafonova
St. Petersburg State University
Author for correspondence.
Email: ivagafonovaspb@gmail.com
Russian Federation, Universitetskaya nab. 7/9, St. Petersburg, 199034
V. N. Malozemov
St. Petersburg State University
Email: ivagafonovaspb@gmail.com
Russian Federation, Universitetskaya nab. 7/9, St. Petersburg, 199034