# Quasipower Series And Quasianalytic Classes Of Functions (translations Of Mathematical Monographs)

by G. V. Badalyan /
2002 / English / PDF, DjVu

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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.

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.