TY - JOUR

T1 - Extremal polynomials related to Zolotarev polynomials

AU - Agafonova, I.V.

AU - Malozemov, V.N.

PY - 2016

Y1 - 2016

N2 - © 2016, Pleiades Publishing, Ltd.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.

AB - © 2016, Pleiades Publishing, Ltd.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.

U2 - 10.1134/S1064562416020113

DO - 10.1134/S1064562416020113

M3 - Article

VL - 93

SP - 164

EP - 165

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - 2

ER -