Asymptotics of polynomials orthogonal with respect to varying measures

Abstract

Letdσ be a finite positive Borel measure on the interval [0, 2π] such that σ′>0 almost everywhere; andW n be a sequence of polynomials, degW n =n, whose zeros (w n ,1,⋯,w n,n lie in [|z|≤1]. Let ∥dσ n ∥<+∞ for eachn∈N, wheredσ n =dσ/|W n (e iθ )|2. We consider the table of polynomialsϕ n,m such that for each fixedn∈N the systemϕ n,m,m∈N, is orthonormal with respect todσ n . If $$\mathop {\lim }\limits_n \sum\limits_{\iota = 1}^n {(1 - |w_{n,\iota } |) = + \infty }$$ andk∈N then lim n ϕ n,n+k+1(w)/ϕ n,n+k (w)=w uniformly on each compact set contained in [|w|≥1]. This result extends a well-known one of E. A. Rakhmanov. Extensions of several results of A. Mate, P. Nevai, and V. Totik are also obtained; e.g., the above conditions also yield $$\mathop {\lim }\limits_n \int {(\sqrt {\sigma '(\theta )} } |\varphi _{n,n + k} (e^{i\theta } )/W_n (e^{i\theta } )| - 1)^2 d\theta = 0$$ which enables us to restate much of Szego's theory in this new setting. Weak convergence results of orthogonal polynomials on the real line are also obtained.