Quasipower Series and Quasianalytic Classes of Functions
About this Title
G. V. Badalyan, Armenian Academy of Sciences, Yerevan, Armenia. Translated by D. M. Chibisov
Publication: Translations of Mathematical Monographs
Publication Year: 2002; Volume 216
ISBNs: 978-0-8218-2943-1 (print); 978-1-4704-4641-3 (online)
MathSciNet review: MR1937853
MSC: Primary 30D60; Secondary 26E10, 30B10, 30D35, 40G10
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.
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.
Research mathematicians and advanced graduate students.
Table of Contents
- Quasianalytic classes of functions
- Generalizations of the Taylor formula. Quasipower series
- Functions of Carleman’s classes: Expansion in quasipower series
- Criteria for the possibility of expanding functions in quasipower and factorial series
- Generalized completely monotone functions and the condition for absolute convergence of a quasipower series (in the basic interval)
- On the use of quasipower series for representation of analytic functions in non-circular domains
- Some applications of quasipower series to the theory of functions of quasianalytic classes