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Topics in Interpolation Theory of Rational Matrix-valued Functions

  • I. Gohberg

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

Table of contents

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

About this book

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

  • DOI https://doi.org/10.1007/978-3-0348-5469-6
  • Copyright Information Birkhäuser Basel 1988
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-5471-9
  • Online ISBN 978-3-0348-5469-6
  • Series Print ISSN 0255-0156
  • Series Online ISSN 2296-4878
  • Buy this book on publisher's site
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