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
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. Maté, P. Nevai, and V. Totik are also obtained; e.g., the above conditions also yield
which enables us to restate much of Szegö's theory in this new setting. Weak convergence results of orthogonal polynomials on the real line are also obtained.
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Communicated by Edward Saff.
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Lagomasino, G.L. Asymptotics of polynomials orthogonal with respect to varying measures. Constr. Approx 5, 199–219 (1989). https://doi.org/10.1007/BF01889607
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DOI: https://doi.org/10.1007/BF01889607