Sharp lower bounds for the Widom factors on the real line

Abstract

We derive lower bounds for the Lp(μ) norms of monic extremal polynomials with respect to compactly supported probability measures μ. We obtain a sharp universal lower bound for all 0<p<∞ and all measures in the Szegő class and an improved lower bound on L2(μ) 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 R.