Extremal polynomials connected with Zolotarev polynomials

Abstract

Let two points a and b be given on the real axis, located to the right and left of the segment [−1, 1] respectively. The extremal problem is posed: find an algebraic polynomial of n-th degree, which at the point a takes value A, on the segment [−1, 1] does not exceed M in modulus and takes the largest possible value at b. This problem is related to the second problem of Zolotarev. In the article the set of values of the parameter A for which this problem has a unique solution is indicated, and an alternance characteristic of this solution is given. The behavior of the solution with respect to the parameter A is studied. It turns out that for some A the solution can be obtained with the help of the Chebyshev polynomial, while for all other admissible A with the help of the Zolotarev polynomial.