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The second volume of the two volumes book is dedicated to various
extensions and generalizations of Dyadic (Walsh) analysis and
related applications. Considered are dyadic derivatives on Vilenkin
groups and various other Abelian and finite non-Abelian groups.
Since some important results were developed in former Soviet Union
and China, we provide overviews of former work in these countries.
Further, we present translations of three papers that were
initially published in Chinese. The presentation continues with
chapters written by experts in the area presenting discussions of
applications of these results in specific tasks in the area of
signal processing and system theory. Efficient computing of related
differential operators on contemporary hardware, including graphics
processing units, is also considered, which makes the methods and
techniques of dyadic analysis and generalizations computationally
feasible. The volume 2 of the book ends with a chapter presenting
open problems pointed out by several experts in the area.
Dyadic (Walsh) analysis emerged as a new research area in applied
mathematics and engineering in early seventies within attempts to
provide answers to demands from practice related to application of
spectral analysis of different classes of signals, including audio,
video, sonar, and radar signals. In the meantime, it evolved in a
mature mathematical discipline with fundamental results and
important features providing basis for various applications. The
book will provide fundamentals of the area through reprinting
carefully selected earlier publications followed by overview of
recent results concerning particular subjects in the area written
by experts, most of them being founders of the field, and some of
their followers. In this way, this first volume of the two volume
book offers a rather complete coverage of the development of dyadic
Walsh analysis, and provides a deep insight into its mathematical
foundations necessary for consideration of generalizations and
applications that are the subject of the second volume. The
presented theory is quite sufficient to be a basis for further
research in the subject area as well as to be applied in solving
certain new problems or improving existing solutions for tasks in
the areas which motivated development of the dyadic analysis.
In recent years important progress has been made in the study of
semi-groups of operators from the viewpoint of approximation
theory. These advances have primarily been achieved by introducing
the theory of intermediate spaces. The applications of the theory
not only permit integration of a series of diverse questions from
many domains of mathematical analysis but also lead to significant
new results on classical approximation theory, on the initial and
boundary behavior of solutions of partial differential equations,
and on the theory of singular integrals. The aim of this book is to
present a systematic treatment of semi groups of bounded linear
operators on Banach spaces and their connec tions with
approximation theoretical questions in a more classical setting as
well as within the setting of the theory of intermediate spaces.
However, no attempt is made to present an exhaustive account of the
theory of semi-groups of operators per se, which is the central
theme of the monumental treatise by HILLE and PHILLIPS (1957).
Neither has it been attempted to give an account of the theory of
approximation as such. A number of excellent books on various
aspects of the latter theory has appeared in recent years, so for
example CHENEY (1966), DAVIS (1963), LORENTZ (1966), MEINARDUS
(1964), RICE (1964), SARD (1963). By contrast, the present book is
primarily concerned with those aspects of semi-group theory that
are connected in some way or other with approximation."
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