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Extremal Polynomials Connected with Zolotarev Polynomials

Abstract Let two points a and b located to the right and left of the interval [–1, 1], respectively, be given on the real axis. The extremal problem is stated as follows: find an algebraic polynomial of the n -th degree, whose value is A at point a , it does not exceed M in absolute value in the interval [–1, 1], and takes the largest possible value at point b . This problem is connected with the second Zolotarev problem. A set of values of the parameter A , for which this problem has a unique solution, is indicated in this paper and an alternance characteristic of this solution is given. The behavior of the solution with respect to the parameter A is studied. It is found that the solution can be obtained for certain A using the Chebyshev polynomial, and can be obtained for all other admissible A with the help of the Zolotarev polynomial.