Abstract. Let $\mu $ be a positive measure on $[-1,1]$ satisfying the Szegő condition $$ \int_{-1}^1 {\log\mu ^\prime(t) \over\sqrt {1-t^2}} dt > -\infty $$ with $\mu ^\prime $ being the Radon--Nikodym derivative with respect to the Lebesgue measure, and let further $w_{2n} \in{\cal P}_{2n}$, $n\in{\msbm N}$, be a sequence of polynomials of degree at most $2 n$, with zeros $a_{2n,1},\ldots,a_{2n,m} \in{\msbm C}\setminus[-1,1]$, $m = \deg(w_{2n})$, and being positive on $[-1,1]$. We study the asymptotic behavior of the polynomials $p_n = p_n(\mu_n; \cdot )$ orthonormal with respect to the varying measure $d\mu_n := w_{2n}^{-1} d\mu $. Let $\varphi\colon \overline{\msbm C}\setminus[-1,1] \rightarrow{\msbm D}$ be the conformal mapping of $\overline{\msbm C}\setminus[-1,1]$ onto ${\msbm D}$ with $\varphi(\infty ) = 0$ and $\varphi ^\prime(\infty ) > 0$. Under the assumption that $$(\ast)\qquad\qquad\lim _{n \to\infty } \left[ (2n - \deg(w_{2n})) + \sum_{j = 1}^{\deg(w_{2n})} (1 - | \varphi(a_{2n,j})| ) \right ] = \infty $$ we prove strong asymptotic relations in ${\msbm C}\setminus[-1,1]$ for the orthonormal polynomials $p_n (\mu_n; \cdot )$ as $n \to\infty $. An analogue of this result is proved for polynomials that are orthonormal with respect to varying measures on the unit circle ${\msbm T}= \partial{\msbm D}$. Also in this case the original measure of orthogonality has to belong to the Szegő class (on ${\msbm T}$) and an assumption analoguous to (*) has to hold true. The necessity of this assumption will be discussed in some detail for the case of orthogonality on ${\msbm T}$. The strong asymptotic relations for $p_n (\mu_n;\cdot )$ lead to correspondingly precise asymptotic error estimates for multipoint Padé approximants (i.e., rational interpolants) to Markov functions $f(z) = \int(t-z)^{-1} d\mu(t)$. The error estimates for multipoint Padé approximants have been the primary motivation for studying strong asymptotics in the present paper.

AMS Subject Classification (1991): 42C05, 41A25

Keyword(s): Orthonormal polynomials with varying weights, Szegő asymptotics, strong asymptotics, multipoint Padé approximants, Markov functions, Markov's Theorem

Received February 15, 1999. (Registered under 2727/2009.)