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Finite Blaschke Products and Their Connections

  • Stephan Ramon Garcia
  • Javad Mashreghi
  • William T. Ross

Table of contents

  1. Front Matter
    Pages i-xix
  2. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 1-19
  3. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 21-38
  4. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 39-58
  5. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 59-73
  6. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 75-99
  7. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 101-128
  8. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 129-153
  9. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 155-180
  10. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 181-207
  11. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 209-244
  12. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 245-259
  13. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 261-290
  14. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 291-306
  15. Back Matter
    Pages 307-328

About this book

Introduction

This monograph offers an introduction to finite Blaschke products and their connections to complex analysis, linear algebra, operator theory, matrix analysis, and other fields.  Old favorites such as the Carathéodory approximation and the Pick interpolation theorems are featured, as are many topics that have never received a modern treatment, such as the Bohr radius and Ritt's theorem on decomposability.  Deep connections to hyperbolic geometry are explored, as are the mapping properties, zeros, residues, and critical points of finite Blaschke products.  In addition, model spaces, rational functions with real boundary values, spectral mapping properties of the numerical range, and the Darlington synthesis problem from electrical engineering are also covered.

Topics are carefully discussed, and numerous examples and illustrations highlight crucial ideas. While thorough explanations allow the reader to appreciate the beauty of the subject, relevant exercises following each chapter improve technical fluency with the material.  With much of the material previously scattered throughout mathematical history, this book presents a cohesive, comprehensive and modern exposition accessible to undergraduate students, graduate students, and researchers who have familiarity with complex analysis.

Keywords

Blaschke Products Mobius Transformations Polynomials Operator Theory Hilbert Spaces Complex Analysis Functional Analysis Finite Blaschke products Gauss-Lucas theorem Lagrange interpolation Group Theory real complex functions Darlington synthesis problem finite-dimensional model spaces

Authors and affiliations

  • Stephan Ramon Garcia
    • 1
  • Javad Mashreghi
    • 2
  • William T. Ross
    • 3
  1. 1.Department of MathematicsPomona CollegeClaremontUSA
  2. 2.Department of Mathematics and StatisticsLaval UniversityQuébec CityCanada
  3. 3.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-78247-8
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-78246-1
  • Online ISBN 978-3-319-78247-8
  • Buy this book on publisher's site