# Sharp lower bounds for the Widom factors on the real line

## TL;DR

This paper establishes sharp lower bounds for the Widom factors related to extremal polynomials on the real line, applicable across various classes of measures and orthogonal polynomials, advancing understanding of polynomial norm behavior.

## Contribution

It provides the first universal lower bounds for Widom factors on the real line, including improved bounds for specific classes of orthogonal polynomials.

## Key findings

- Universal lower bound for all $0<p<
$ and measures in the Szeg\
- Improved lower bounds for $L^2(\mu)$ norms for Jacobi and other orthogonal polynomials
- Bounds are sharp and applicable to a wide range of measures and polynomial classes.

## Abstract

We derive lower bounds for the $L^p(\mu)$ norms of monic extremal polynomials with respect to compactly supported probability measures $\mu$. We obtain a sharp universal lower bound for all $0<p<\infty$ and all measures in the Szeg\H{o} class and an improved lower bound on $L^2(\mu)$ norm for several classes of orthogonal polynomials including Jacobi polynomials, isospectral torus of a finite gap set and orthogonal polynomials with respect to the equilibrium measure of an arbitrary non-polar compact subset of $\mathbb{R}$.