Extremal problems of Chebyshev type

Abstract

Let $a \in \mathbb {C} \setminus [-1,1]$ be given. We consider the problem of finding $\sup |p(a)|$ among all polynomials $p$ with complex coefficients of degree less than or equal to $n$ with $\max _{-1\leq x \leq 1}|p(x)| \leq 1$. We derive an asymptotic expression for the extremal polynomial and for the extremal value in terms of elementary functions. The solution is based on the description of Zolotarev polynomials with respect to square root polynomial weights.

Locations

• Proceedings of the American Mathematical Society - View - PDF