**Translations of Mathematical Monographs**

2002;
183 pp;
Hardcover

MSC: Primary 30;

**Print ISBN: 978-0-8218-2943-1
Product Code: MMONO/216**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

**Electronic ISBN: 978-1-4704-4641-3
Product Code: MMONO/216.E**

List Price: $88.00

AMS Member Price: $70.40

MAA Member Price: $79.20

# Quasipower Series and Quasianalytic Classes of Functions

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*G. V. Badalyan*

In this book, G. V. Badalyan addresses the fundamental problems of the
theory of infinitely-differentiable functions using the theory of
functions of quasianalytic classes.

A certain class of functions \(C\) on an interval is called
quasianalytic if any function in \(C\) is uniquely determined by the
values of its derivatives at any point. The obvious question, then, is how to
reconstruct such a function from the sequence of values of its derivatives at a
certain point. In order to answer that question, Badalyan combines a study of
expanding functions in generalized factorial series with a study of quasipower
series.

The theory of quasipower series and its application to the
reconstruction problem are explained in detail in this research
monograph. Along the way other, related problems are solved, such as
Borel's hypothesis that no quasianalytic function can have all positive
derivatives at a point.

Originally published in Russian, this English translation contains
additional material that treats the problems of classification of
infinitely-differentiable functions, conditions for absolute convergence
of quasipower series in terms of the functions that generate them, and
the possibility of representing analytic functions by quasipower series
in non-circular domains.

While the treatment is technical, the theory is developed chapter by
chapter in detail, and the first chapter is of an introductory nature.
The quasipower series technique explained here provides the means to
extend the previously known results and elucidates their nature in
the most relevant manner. This method also allows for thorough
investigation of numerous problems of the theory of functions of
quasianalytic classes by graduate students and research mathematicians.

#### Readership

Research mathematicians and advanced graduate students.

#### Table of Contents

# Table of Contents

## Quasipower Series and Quasianalytic Classes of Functions

- Cover Cover11
- Title page v6
- Contents vii8
- Preface ix10
- Introduction xi12
- Quasianalytic classes of functions 118
- Generalizations of the Taylor formula. Quasipower series 2744
- Functions of Carleman’s classes: Expansion in quasipower series 6380
- Criteria for the possibility of expanding functions in quasipower and factorial series 89106
- Generalized completely monotone functions and the condition for absolute convergence of a quasipower series (in the basic interval) 119136
- On the use of quasipower series for representation of analytic functions in non-circular domains 139156
- Some applications of quasipower series to the theory of functions of quasianalytic classes 153170
- Bibliography 181198
- Back Cover Back Cover1201