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Asymptotics of polynomials orthogonal with respect to varying measures

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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. Maté, 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 Szegö's theory in this new setting. Weak convergence results of orthogonal polynomials on the real line are also obtained.

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Authors and Affiliations

  1. Havana University, Havana, Cuba

    Guillermo López Lagomasino

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  1. Guillermo López Lagomasino
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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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