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Finite Blaschke Products and Their Connections

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  • © 2018

Overview

Authors:
  1. Stephan Ramon Garcia

    1. Department of Mathematics, Pomona College, Claremont, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Javad Mashreghi

    1. Department of Mathematics and Statistics, Laval University, Québec City, Canada

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    You can also search for this author in PubMed   Google Scholar

  3. William T. Ross

    1. Department of Mathematics and Computer Science, University of Richmond, Richmond, USA

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  • Explains connections in complex analysis, linear algebra, operator theory, and electrical engineering

  • Accessible to undergraduate students, graduate students, and researchers familiar with complex analysis

  • Offers numerous exercises that equip the reader with a working knowledge of the material

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. Geometry of the Schur Class

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 1-19

  3. Elementary Hyperbolic Geometry

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 21-38

  4. Finite Blaschke Products: The Basics

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 39-58

  5. Approximation by Finite Blaschke Products

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 59-73

  6. Zeros and Residues

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 75-99

  7. Critical Points

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 101-128

  8. Interpolation

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 129-153

  9. The Bohr Radius

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 155-180

  10. Finite Blaschke Products and Group Theory

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 181-207

  11. Finite Blaschke Products and Operator Theory

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 209-244

  12. Real Complex Functions

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 245-259

  13. Finite-Dimensional Model Spaces

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 261-290

  14. The Darlington Synthesis Problem

    • Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
    Pages 291-306

  15. Back Matter

    Pages 307-328
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Keywords

About this book

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.


Reviews

“The book under consideration is concerned with finite Blaschke products. … The book is designed for students and researchers familiar with basic real and complex analysis and linear algebra. The proofs are detailed and dozens illustrations are provided. At the end of each chapter, the authors include exercises so that the reader can gain greater technical fluency with the material and appreciate the beauty of the subject.” (Leonid Golinskii, zbMATH 1398.30002, 2018)

Authors and Affiliations

  • Department of Mathematics, Pomona College, Claremont, USA

    Stephan Ramon Garcia

  • Department of Mathematics and Statistics, Laval University, Québec City, Canada

    Javad Mashreghi

  • Department of Mathematics and Computer Science, University of Richmond, Richmond, USA

    William T. Ross

About the authors

Stephan Ramon Garcia is a professor of mathematics at Pomona College



Javad Mashreghi is a professor in the Department of Mathematics and Statistics at Laval University, Alexandre-Vachon. 




William T. Ross is a professor of mathematics and the Chair of the Department of Math and Computer Sciences at the University of Richmond.




Bibliographic Information

  • Book Title: Finite Blaschke Products and Their Connections

  • Authors: Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

  • DOI: https://doi.org/10.1007/978-3-319-78247-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing AG, part of Springer Nature 2018

  • Hardcover ISBN: 978-3-319-78246-1Published: 05 June 2018

  • Softcover ISBN: 978-3-030-08655-8Published: 19 December 2018

  • eBook ISBN: 978-3-319-78247-8Published: 24 May 2018

  • Edition Number: 1

  • Number of Pages: XIX, 328

  • Number of Illustrations: 39 b/w illustrations, 10 illustrations in colour

  • Topics: Functional Analysis, Operator Theory

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