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previously unpublished method of “continuation” to the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.