Finite Blaschke product interpolation on the closed unit disc
type: Item
instance of: scholarly article
title: Finite Blaschke product interpolation on the closed unit disc
zbMATH Open document ID: 1014.30026
DOI: 10.1016/S0022-247X(02)00249-4
author: Christer Glader
author: Mikael Lindström
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: Let \(\{z_j\}_{j=1}^n\) be distinct points and \(\{w_j\}_{j=1}^n\) be complex numbers such that \(|z_j|= 1\), \(|w_j|=1\), \(j =1,\dots,n_1\), \((n_1\geqslant 1)\), and \(|z_j|< 1\), \(|w_j|< 1\), \(j = n_1+1,\dots,n_1 + n_2=n\). Suppose that the Pick matrix \(Q_{n_2}\) defined by \(Q_{ij}=(1-w_i\overline{w}_j)/(1-z_i\overline{z}_j)\), \(i,j = n_1+1, \dots,n\) is positive semi-definite and of full rank. Based on a result of W.~B.~Jones and S.~Ruscheweyh the authors show that there exists a finite Blaschke product \(B(z)\) of degree at most \(n-1\) that satisfies \[ B(z_j) =w_j,\quad 1\leqslant j\leqslant n.\tag{1} \] A Blaschke product interpolation problem of this kind occurs in the design of digital filters. All finite Blaschke products \(B_n(z)\) of degree \(n\) in the form \[ \frac{\alpha_0+\alpha_1z+\dots+\alpha_nz^n}{\overline{\alpha}_0+ \overline{\alpha}_1z+\dots+\overline{\alpha}_nz^n},\quad\alpha_j\in\mathbb C, \quad j=0,\dots,n, \tag{2} \] that satisfy (1) for given Nevanlinna-Pick data \(\{z_j\}_{j=1}^n\), \(\{w_j\}_{j=1}^n\) in the open unit disc \(D\) with the positive semi-definite Pick matrix \(Q_n\), and the scaled finite Blaschke product \(cB_m(z)\) with \(m\leqslant n-1\) of minimal norm on \(D\), that satisfies (1) for the data \(\{z_j\}_{j=1}^n\subset D\) and \(\{w_j\}_{j=1}^n\subset\mathbb C\), are constructed by solving eigenvalue problems of the interpolation data. Finally, a numerical method for determining an interpolating finite Blaschke product of degree at most \(n-1\) is discussed.
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30-06
Mathematics Subject Classification ID: 62M20
zbMATH DE Number: 1866576
zbMATH Keywords: Nevanlinna-Pick interpolation
zbMATH Keywords: Nevanlinna-Pick data
zbMATH Keywords: interpolating finite Blaschke product
zbMATH Keywords: semi-definite Pick matrix
zbMATH Keywords: eigenvalue problems
zbMATH Keywords: digital filters
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
cites work: A Note on Bounded Interpolation in the Unit Disc
cites work: Q3922907
cites work: Blaschke product interpolation and its application to the design of digital filters
cites work: Shorter Notes: Interpolation by a Finite Blaschke Product
description: scientific article; zbMATH DE number 1866576