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 $\alpha +\beta \ge 1$, $\alpha ,\beta \in \mathbb{N}\cup \{0\}$ 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 $[W_{2,n}(\mu ){]}^2\ge 2S(\mu )$ 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.