Finite Blaschke Products and Their Connections
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 interp
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Blaschke Products and Their Connections
Finite Blaschke Products and Their Connections
Stephan Ramon Garcia • Javad Mashreghi William T. Ross
Finite Blaschke Products and Their Connections
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Stephan Ramon Garcia Department of Mathematics Pomona College Claremont, CA, USA
Javad Mashreghi Department of Mathematics and Statistics Laval University Québec City, QC, Canada
William T. Ross Department of Mathematics and Computer Science University of Richmond Richmond, VA, USA
ISBN 978-3-319-78246-1 ISBN 978-3-319-78247-8 (eBook) https://doi.org/10.1007/978-3-319-78247-8 Library of Congress Control Number: 2018937693 Mathematics Subject Classification (2010): 30J10, 30J05, 30A78, 30A76, 30A08, 30C15, 30E10, 47A12, 47N50, 47A30 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To our families: Gizem; Reyhan, and Altay Shahzad; Dorsa, Parisa, and Golsa Fiona
Preface
This is a book about a beautiful subject that begins with the topic of Möbius transformations. Indeed, Möbius transformations z →
az + b cz + d
are studied in complex analysis since their mapping properties demonstrate wonderful connections with geometry. These transformations map extended circles to extended circles, enjoy the symmetry principle, come in several types yielding different behavior depending on their fixed point(s), and, through an identification with 2 × 2 matrices, make connections to group theory and projective geometry. Finite Blaschke products, the focus of this book, are products of certain types of Möbius transformations, the automorphisms of the open unit disk D, namely z → ξ
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