Asymptotics for minimal Blaschke products and best \(L_1\) meromorphic approximants of Markov functions.
type: Item
instance of: scholarly article
title: Asymptotics for minimal Blaschke products and best \(L_1\) meromorphic approximants of Markov functions.
zbMATH Open document ID: 1038.41005
DOI: 10.1007/BF03321004
author: L. Baratchart
author: Vasiliy A. Prokhorov
author: E. B. Saff
published in: Computational Methods and Function Theory
publication date: 2003-01-26T00:00:00Z
review text: 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 \(G\setminus [a,b]\).
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 1851122
zbMATH Keywords: Blaschke
zbMATH Keywords: meromorphic approximation
zbMATH Keywords: Markov function
zbMATH Keywords: best approximation
reviewed by: Aldric L. Brown
MaRDI profile type: Publication
cites work: Best rational approximation to Markov functions
cites work: Best meromorphic approximation of Markov functions on the unit circle.
cites work: Q4534547
cites work: Asymptotic error for \(L^2\) best rational approximants to Markov functions
cites work: Q5618011
cites work: The n-width of sets of analytic functions
cites work: Q4750081
cites work: The asymptotic distribution of zeros of minimal Blaschke products
cites work: Q3922907
cites work: Q3874458
cites work: Q5289122
cites work: WIDTHS OF A CLASS OF ANALYTIC FUNCTIONS
cites work: Singular numbers of the imbedding operators for certain classes of analytic and harmonic functions
cites work: ASYMPTOTICS OF SINGULAR NUMBERS OF IMBEDDING OPERATORS FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS
cites work: Q3344608
cites work: Q4328333
cites work: The 𝐻^{𝑝} spaces of an annulus
cites work: Q4360405
cites work: Q5289656
cites work: Weighted approximation with varying weight
cites work: Inner and Outer Functions on Riemann Surfaces
Recommended article: Best meromorphic approximation of Markov functions on the unit circle.
Recommended article: Non-uniqueness of rational best approximants
Recommended article: Growth and zero sets of analytic families of Cauchy-Stieltjes integrals
Recommended article: Zero distribution and factorization of analytic functions of slow growth in the unit disc
Recommended article: On the Maximum and Minimum Modulus of Rational Functions
Recommended article: An Extreme Absolutely Continuous RP-Measure
Recommended article: Bounded point evaluations for nonminimal measures
Recommended article: Majorants of meromophic functions with fixed poles
Recommended article: Dominating measures for spaces of analytic functions
Recommended article: ASYMPTOTICS OF Lp EXTREMAL POLYNOMIALS OFF THE UNIT CIRCLE
description: scientific article; zbMATH DE number 1851122