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.
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References
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Original Russian Text © I.V. Agafonova, V.N. Malozemov, 2016, published in Doklady Akademii Nauk, 2016, Vol. 467, No. 3, pp. 255–256.
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Agafonova, I.V., Malozemov, V.N. Extremal polynomials related to Zolotarev polynomials. Dokl. Math. 93, 164–165 (2016). https://doi.org/10.1134/S1064562416020113
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DOI: https://doi.org/10.1134/S1064562416020113