A lower bound for the norm of the minimal residual polynomial

TL;DR

This paper establishes a new lower bound for the norm of minimal residual polynomials on certain complex sets, refining existing inequalities and contributing to approximation theory.

Contribution

It derives a sharper inequality for the norm of minimal residual polynomials on finite unions of real intervals, improving the Bernstein--Walsh Lemma.

Findings

New lower bound for polynomial norms on complex sets

Refinement of Bernstein--Walsh Lemma

Limit of the polynomial norm sequence exists

Abstract

Let $S$ be a compact infinite set in the complex plane with $0\notin{S}$, and let $R_n$ be the minimal residual polynomial on $S$, i.e., the minimal polynomial of degree at most $n$ on $S$ with respect to the supremum norm provided that $R_n(0)=1$. For the norm $L_n(S)$ of the minimal residual polynomial, the limit $\kappa(S):=\lim_{n\to\infty}\sqrt[n]{L_n(S)}$ exists. In addition to the well-known and widely referenced inequality $L_n(S)\geq\kappa(S)^n$, we derive the sharper inequality $L_n(S)\geq2\kappa(S)^n/(1+\kappa(S)^{2n})$ in the case that $S$ is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein--Walsh Lemma.