Bounds on Orthonormal Polynomials for Restricted Measures

Abstract

Suppose that \(\nu \) is a given positive measure on \(\left[ -1,1\right] \), and that \(\mu \) is another measure on the real line, whose restriction to \( \left( -1,1\right) \) is \(\nu \). We show that one can bound the orthonormal polynomials \(p_{n}\left( \mu ,y\right) \) for \(\mu \) and \(y\in \mathbb {R}\), by the supremum of \(\left| S_{J}\left( y\right) p_{n-J}\left( S_{J}^{2}\nu ,y\right) \right| \), where the sup is taken over all \(0\le J\le n\) and all monic polynomials \(S_{J}\) of degree J with zeros in an appropriate set.