Asymptotics for minimal Blaschke products and best \(L_1\) meromorphic approximants of Markov functions.
The `minimal' and `best' of the title are related in the following way. Let \(\mu\) be a positive Borel measure with infinite support \(E\subset (-1,1)\). Let \({\mathcal B}_n\) be the collection of all Blaschke products of degree \(n\). The Markov function defined by \(\mu\) is \[ f(z)= {1\over 2\pi i}\int {d\mu(x)\over z- x}. \] \({\mathcal M}_{n,1}(G)\), where \(G\) is the open unit disc of the complex plane, is the set of all meromorphic functions on \(G\) of the form \(h= P/Q\) where \(P\) is in the Hardy space \(H_1(G)\) and \(Q\) is a polynomial of degree at most \(n\). It is stated, with reference to \textit{J.-E. Andersson} [J. Approximation Theory 76, 219--232 (1994; Zbl 0796.41014)], that \[ \Delta_n:= \inf_{B\in{\mathcal B}_n} \int_E| B(x)|^2 \,d\mu(x)= \inf_{h\in{\mathcal M}_{n,1}(G)}\| f- h\|_1, \] and there exist \(B_n\in{\mathcal B}_n\) and \(h_n\in{\mathcal M}_{n,1}(G)\) for which the two infima are attained and \(h_nB_n\in H_1(G)\). The present paper considers \(\mu\) with support an interval \(E= [a,b]\) and for which the Szegő condition \[ \int^b_a {\log(d\mu/dx)\over \sqrt{(x- a)(x-b)}}\,dx> -\infty \] is satisfied. It is a sequel to the same authors' paper [Found. Comput. Math. 1, 385--416 (2001; Zbl 1053.41019)] and is concerned with the sequence of measures \((B^2_n/\Delta_n)\) (Theorem 4 states that it is a weak\(^*\) convergent and identifies the limit) and the asymptotic behaviour of the sequence of functions \((f- h_n)(z)\) for \(z\in \mathbb{C}\setminus(E\cup E^{-1})\). An appendix includes a detailed treatment of a factorization theorem (which is generalizable) for Hardy spaces on the slit disc \(G\setminus [a,b]\).
scientific article; zbMATH DE number 1851122
scientific article