Widom factors for Chebyshev and residual polynomials on semi-regular subsets of $\mathbb{R}$

Abstract

We study $L^\infty$ Widom factors for Chebyshev and residual polynomials on compact non-polar subsets of the real line that need not be regular in the sense of potential theory and, in particular, on sets with isolated points. We first show that boundedness of the Widom factors is independent of the normalization point $x_*\in\overline{\mathbb{R}}\backslash\mathsf{E}$. For semi-regular sets, meaning that the set of regular points $\mathsf{E}^\mathrm{reg}$ is closed, we introduce the irregularity coefficient $\mathcal{IR}(\mathsf{E},x_*)=\sum_{x\in\mathsf{E}\backslash\mathsf{E}^\mathrm{reg}}G_\mathsf{E}(x,x_*)$, which measures the contribution of irregular boundary points to the extremal polynomial problem. Our upper bound is \[ \sup_{n\ge1}\mathcal{W}_n(\mathsf{E},x_*) \le 2\exp\bigl[\mathcal{PW}(\mathsf{E}^\mathrm{reg},x_*)+\mathcal{IR}(\mathsf{E},x_*)\bigr], \] while the complementary lower bound is \[ \liminf_{n\to\infty}\mathcal{W}_n(\mathsf{E},x_*) \ge 2\exp\bigl[\mathcal{IR}(\mathsf{E},x_*)\bigr]. \] Consequently, for semi-regular Parreau--Widom sets the Widom factors are bounded if and only if $\mathcal{IR}(\mathsf{E},x_*)<\infty$. When the regular part is a finite union of intervals, this condition is equivalent to a geometric square-root summability condition on the irregular points. In the special case $\mathsf{E}=[a,b]\cup\{x_k\}_{k\ge1}$, the Widom factors have the exact, possibly infinite, limit $2\exp[\mathcal{IR}(\mathsf{E},x_*)]$. When this limit is finite, we obtain the corresponding Szegő--Widom asymptotics for the normalized extremal polynomials.