Computational Methods and Function Theory 1 (2001), No. 2, 501--520
Copyright Heldermann Verlag 2001
Laurent Baratchart
baratcha@sophia.inria.fr, INRIA, 2004 Route des Lucioles B.P.93, 06902 Sophia Antipolis Cedex, France.
Vasiliy A. Prokhorov
prokhorov@mathstat.usouthal.edu, Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688-0002, U.S.A.
Edward B. Saff
esaff@math.vanderbilt.edu, Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240, U.S.A.
Let $\mu$ be a positive Borel measure with support $\supp \mu =E\subset(-1,1)$ and let $$ \Delta_{n}= \inf_{B\in\mathcal{B}_n}\int_E|B(x)|^2d \mu(x), $$ where $\mathcal{B}_n$ is the collection of all Blaschke products of degree $n$. Denote by $ B_n \in \mathcal{B}_n$ a Blaschke product that attains the value $\Delta_n$. We investigate the asymptotic behavior, as $n \to \infty$, of the minimal Blaschke products $B_n$ in the case when the measure $\mu$ with support $E=[a,b]$ satisfies the Szeg\H{o} condition: $$ \int_a^b \frac{\log(d \mu/ d x)}{\sqrt{(x-a)(b-x)}} \,dx > - \infty. $$ At the same time, we shall obtain results related to the convergence of best $L_1$ approximants on the unit circle to the Markov function $$ f(z)= \frac{1}{2 \pi i} \int_E\frac{d \mu(x)}{z-x} $$ by meromorphic functions of the form $P/Q$, where $P$ belongs to the Hardy space $H_1$ of the unit disk and $Q$ is a polynomial of degree at most $n$. We also include in an appendix a detailed treatment of a factorization theorem for Hardy spaces of the slit disk, which may be of independent interest.
Keywords: Blaschke products, meromorphic approximation, Markov functions, best approximation.
MSC 2000: 41A20, 30E10, 47B35.
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