A remark on the representation of Zolotarev polynomials

Abstract

Zolotarev polynomials are the polynomials that have minimal deviation from zero on [−1, 1] with respect to the norm ‖x n − σ x n−1 + a n−2 x n−2 + … + a 1 x + a n ‖ for given σ and for all a k ∈ ℛ.This note complements the paper of F. Pehersforfer [J. London Math. Soc. (1) 74 (2006) 143-153] with exact (not asymptotic) construction of the Zolotarev polynomials with respect to the norm L 1 for |σ| < 1 and with respect to the norm L 2 for |σ| ≠ 1 in the form of Bernstein-Szego orthogonal polynomials. For all σ ∈ ℛ in L 1 and L 2 norms, the Zolotarev polynomials satisfy exactly (not asymptotically) the triple recurrence relation of the Chebyshev polynomials.