# Topics in Interpolation Theory of Rational Matrix-valued Functions

• I. Gohberg
Book

Part of the Operator Theory: Advances and Applications book series (OT, volume 33)

1. Front Matter
Pages I-IX
2. Joseph A. Ball, Israel Gohberg, Leiba Rodman
Pages 1-72
3. I. Gohberg, M. A. Kaashoek, A. C. M. Ran
Pages 73-108
4. I. Gohberg, M. A. Kaashoek
Pages 109-122
5. Joseph A. Ball, Nir Cohen, André C. M. Ran
Pages 123-173
6. Daniel Alpay, Israel Gohberg
Pages 175-222
7. Israel Gohberg, Sorin Rubinstein
Pages 223-247
8. Back Matter
Pages N1-N1

### Introduction

One of the basic interpolation problems from our point of view is the problem of building a scalar rational function if its poles and zeros with their multiplicities are given. If one assurnes that the function does not have a pole or a zero at infinity, the formula which solves this problem is (1) where Zl , " " Z/ are the given zeros with given multiplicates nl, " " n / and Wb" " W are the given p poles with given multiplicities ml, . . . ,m , and a is an arbitrary nonzero number. p An obvious necessary and sufficient condition for solvability of this simplest Interpolation pr- lern is that Zj :f: wk(1~ j ~ 1, 1~ k~ p) and nl +. . . +n/ = ml +. . . +m ' p The second problem of interpolation in which we are interested is to build a rational matrix function via its zeros which on the imaginary line has modulus 1. In the case the function is scalar, the formula which solves this problem is a Blaschke product, namely z z. )mi n u(z) = all = l~ (2) J ( Z+ Zj where [o] = 1, and the zj's are the given zeros with given multiplicities mj. Here the necessary and sufficient condition for existence of such u(z) is that zp :f: - Zq for 1~ ]1, q~ n.

### Keywords

Blaschke product function functions Interpolation Matrix Scala

### Editors and affiliations

• I. Gohberg
• 1
1. 1.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael

### Bibliographic information

• Book Title Topics in Interpolation Theory of Rational Matrix-valued Functions
• Authors I. Gohberg
• Series Title Operator Theory: Advances and Applications
• DOI https://doi.org/10.1007/978-3-0348-5469-6
• Copyright Information Birkhäuser Basel 1988
• Publisher Name Birkhäuser, Basel
• eBook Packages
• Hardcover ISBN 978-3-7643-2233-5
• Softcover ISBN 978-3-0348-5471-9
• eBook ISBN 978-3-0348-5469-6
• Series ISSN 0255-0156
• Series E-ISSN 2296-4878
• Edition Number 1
• Number of Pages IX, 247
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
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