TY - JOUR

T1 - Extremal Polynomials Connected with Zolotarev Polynomials

AU - Agafonova, I. V.

AU - Malozemov, V. N.

N1 - Agafonova, I.V., Malozemov, V.N. Extremal Polynomials Connected with Zolotarev Polynomials. Vestnik St.Petersb. Univ.Math. 53, 1–9 (2020). https://doi.org/10.1134/S1063454120010021

PY - 2020/3/26

Y1 - 2020/3/26

N2 - 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.

AB - 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.

KW - alternance

KW - Chebyshev polynomials

KW - extremal properties of polynomials

KW - Zolotarev polynomials

UR - http://www.scopus.com/inward/record.url?scp=85082627729&partnerID=8YFLogxK

UR - https://link.springer.com/article/10.1134/S1063454120010021

M3 - Article

AN - SCOPUS:85082627729

VL - 53

SP - 1

EP - 9

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -