**University Lecture Series**

Volume: 25;
2002;
149 pp;
Softcover

MSC: Primary 30; 47;

**Print ISBN: 978-0-8218-3175-5
Product Code: ULECT/25**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

**Electronic ISBN: 978-1-4704-2172-4
Product Code: ULECT/25.E**

List Price: $39.00

AMS Member Price: $31.20

MAA Member Price: $35.10

#### Supplemental Materials

# Generalized Analytic Continuation

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*William T. Ross; Harold S. Shapiro*

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previously unpublished method of “continuation” to the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.

#### Readership

Graduate students and research mathematicians interested in functions of a complex variable, approximation theory, and operator theory.

#### Reviews & Endorsements

Interesting and well written book … an extensive and useful bibliography …

-- Zentralblatt MATH

Interesting and inspiring small book … can be recommended as a source of many interesting and well-motivated open problems.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Generalized Analytic Continuation

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Overview 116 free
- Chapter 2. Notation and Preliminaries 1126
- Chapter 3. The Poincaré example 1732
- Chapter 4. Borel's ideas and their later development 2136
- Chapter 5. Gončar Continuation 3752
- Chapter 6. Pseudocontinuation 4560
- 6.1. A problem of Walsh and Tumarkin 4560
- 6.2. Definition and basic examples 4964
- 6.3. Cyclic vectors for the backward shift on H[sup(2)] 5267
- 6.4. The Hardy space of a multiply connected domain 5671
- 6.5. Walsh-Tumarkin again 5873
- 6.6. Other spaces of analytic functions 5974
- 6.7. The Darlington synthesis problem 6378
- 6.8. Linear differential equations of infinite order 6580
- 6.9. Gap theorems 7489
- 6.10. Functional equations and non-continuability 8398

- Chapter 7. A continuation involving almost periodic functions 89104
- Chapter 8. Continuation by formal multiplication of series 99114
- 8.1. Definition 100115
- 8.2. Compatibility with analytic continuation 101116
- 8.3. Cyclic vectors for backward shifts 104119
- 8.4. Spectral properties 113128
- 8.5. Uniqueness 121136
- 8.6. A continuation arising from overconvergence 123138
- 8.7. Generalizing the Walsh-Tumar kin results 125140
- 8.8. Possible generalizations 128143

- Chapter 9. Generalized analytic continuation 131146
- List of Symbols 139154
- Bibliography 141156
- Index 147162
- Back Cover Back Cover1165