Generalized Analytic Continuation
About this Title
William T. Ross, University of Richmond, Richmond, VA and Harold S. Shapiro, Royal Institute of Technology, Stockholm, Sweden
Publication: University Lecture Series
Publication Year 2002: Volume 25
ISBNs: 978-0-8218-3175-5 (print); 978-1-4704-2172-4 (online)
MathSciNet review: MR1895624
MSC: Primary 30B40; Secondary 47A16, 47B38
The theory of generalized analytic continuation studies continuations of meromorphic functions in situations where traditional theory says there is a natural boundary. This broader theory touches on a remarkable array of topics in classical analysis, as described in the book. This book addresses the following questions: (1) When can we say, in some reasonable way, that component functions of a meromorphic function on a disconnected domain, are “continuations” of each other? (2) What role do such “continuations” play in certain aspects of approximation theory and operator theory? The authors use the strong analogy with the summability of divergent series to motivate the subject. In this vein, for instance, theorems can be described as being “Abelian” or “Tauberian”. The introductory overview carefully explains the history and context of the theory.
The authors begin with a review of the works of Poincaré, Borel, Wolff, Walsh, and Gončar, on continuation properties of “Borel series” and other meromorphic functions that are limits of rapidly convergent sequences of rational functions. They then move on to the work of Tumarkin, who looked at the continuation properties of functions in the classical Hardy space of the disk in terms of the concept of “pseudocontinuation”. Tumarkin's work was seen in a different light by Douglas, Shapiro, and Shields in their discovery of a characterization of the cyclic vectors for the backward shift operator on the Hardy space. The authors cover this important concept of “pseudocontinuation” quite thoroughly since it appears in many areas of analysis. They also add a new and previously unpublished method of “continuation” to the list, based on formal multiplication of trigonometric series, which can be used to examine the backward shift operator on many spaces of analytic functions. The book attempts to unify the various types of “continuations” and suggests some interesting open questions.
Graduate students and research mathematicians interested in functions of a complex variable, approximation theory, and operator theory.
Table of Contents
- Chapter 1. Overview
- Chapter 2. Notation and preliminaries
- Chapter 3. The Poincaré example
- Chapter 4. Borel’s ideas and their later development
- Chapter 5. Gončar continuation
- Chapter 6. Pseudocontinuation
- Chapter 7. A continuation involving almost periodic functions
- Chapter 8. Continuation by formal multiplication of series
- Chapter 9. Generalized analytic continuation