Widom factors for generalized Jacobi measures

TL;DR

This paper investigates bounds for Widom factors related to Chebyshev and extremal polynomials on subsets of [-1,1], identifying sets that saturate these bounds and exploring their relationships.

Contribution

It provides new optimal bounds for Widom factors for specific weights and measures, and characterizes sets that achieve these bounds, linking Chebyshev and extremal polynomial properties.

Findings

Identifies sets saturating bounds for Widom factors with specific weights.

Establishes an improved lower bound for Widom factors in certain measures.

Connects saturation conditions between Chebyshev and extremal polynomial Widom factors.

Abstract

We study optimal lower and upper bounds for Widom factors $W_{\infty,n}(K,w)$ associated with Chebyshev polynomials for the weights $w(x)=\sqrt{1+x}$ and $w(x)=\sqrt{1-x}$ on compact subsets of $[-1,1]$. We show which sets saturate these bounds. We consider Widom factors $W_{2,n}(\mu)$ for $L_2(\mu)$ extremal polynomials for measures of the form $d\mu(x)=(1-x)^\alpha (1+x)^\beta d\mu_K(x)$ where $\alpha+\beta\geq 1$, $\alpha,\beta\in\mathbb{N}\cup \{0\}$ and $\mu_K$ is the equilibrium measure of a compact regular set $K$ in $[-1,1]$ with $\pm 1\in K$. We show that for such measures the improved lower bound $[W_{2,n}(\mu)]^2\geq 2S(\mu)$ holds. For the special cases $d\mu(x)=(1-x^2)d\mu_K(x)$, $d\mu(x)=(1-x)d\mu_K(x)$, $d\mu(x)=(1+x)d\mu_K(x)$ we determine which sets saturate this lower bound and discuss how saturated lower bounds for $[W_{2,n}(\mu)]^2$ and $W_{\infty,n}(K,w)$ are related.