Widom factors for generalized Jacobi measures

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.