Unbounded Widom factors for orthogonal and residual polynomials

TL;DR

This paper investigates the unbounded nature of Widom factors for orthogonal and residual polynomials on certain Cantor sets, revealing conditions under which these factors grow without bound and establishing lower bounds based on geometric properties.

Contribution

It introduces constructions of Cantor sets with prescribed Widom factor growth and provides new lower bounds for residual Widom factors depending on geometric configurations.

Findings

Widom factors can be made arbitrarily large on specific Cantor sets.

Residual Widom factors satisfy power-type lower bounds related to gap positions.

Unbounded growth of Widom factors occurs under certain monotonicity and unboundedness conditions.

Abstract

We study Widom factors for (a) monic orthogonal polynomials in $L^2$ with respect to the equilibrium measure of a compact set $K\subset\mathbb{R}$ and (b) residual polynomials normalized at an exterior point. Using weakly equilibrium Cantor sets $K(\gamma)$, we prove: (1) Given any sequence $(c_n)$ with subexponential growth, there exists a non-polar Cantor set $K(\gamma)$, depending on $(c_n)$, such that the $L^2$ Widom factors of the associated orthogonal polynomials (with respect to the equilibrium measure of $K(\gamma)$) exceed $c_n$ for every $n$. (2) For the same $K(\gamma)$ built from $(c_n)$ and each exterior point $x_0\in\mathbb{R}\setminus K(\gamma)$, the residual Widom factors satisfy power-type lower bounds with a Harnack-distance exponent $\tau_{x_0}\in(0,1)$: they are bounded below by $c_n^{\tau_{x_0}}$ for all degrees when $x_0$ lies in an unbounded gap, and along a subsequence of degrees when $x_0$ lies in a bounded gap. Consequently, if, in addition, $(c_n)$ is monotone increasing and unbounded, then the sequence of residual Widom factors is unbounded for every $x_0\in\mathbb{R}\setminus K(\gamma)$. The proofs combine inverse-image constructions, capacity comparisons for period-$n$ sets, harmonic-measure representations for differences of Green functions, and alternation principles on nested approximants.