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Probability theory on compact Lie groups deals with the interaction
between “chance” and “symmetry,” a beautiful area of
mathematics of great interest in its own sake but which is now also
finding increasing applications in statistics and engineering
(particularly with respect to signal processing). The author gives
a comprehensive introduction to some of the principle areas of
study, with an emphasis on applicability. The most important topics
presented are: the study of measures via the non-commutative
Fourier transform, existence and regularity of densities,
properties of random walks and convolution semigroups of measures
and the statistical problem of deconvolution. The emphasis on
compact (rather than general) Lie groups helps readers to get
acquainted with what is widely seen as a difficult field but which
is also justified by the wealth of interesting results at this
level and the importance of these groups for applications. The book
is primarily aimed at researchers working in probability,
stochastic analysis and harmonic analysis on groups. It will also
be of interest to mathematicians working in Lie theory and
physicists, statisticians and engineers who are working on related
applications. A background in first year graduate level measure
theoretic probability and functional analysis is essential; a
background in Lie groups and representation theory is certainly
helpful but the first two chapters also offer orientation in these
subjects.
Probability theory on compact Lie groups deals with the
interaction between chance and symmetry, a beautiful area of
mathematics of great interest in its own sake but which is now also
finding increasing applications in statistics and engineering
(particularly with respect to signal processing). The author gives
a comprehensive introduction to some of the principle areas of
study, with an emphasis on applicability. The most important topics
presented are: the study of measures via the non-commutative
Fourier transform, existence and regularity of densities,
properties of random walks and convolution semigroups of measures
and the statistical problem of deconvolution. The emphasis on
compact (rather than general) Lie groups helps readers to get
acquainted with what is widely seen as a difficult field but which
is also justified by the wealth of interesting results at this
level and the importance of these groups for applications.
The book is primarily aimed at researchers working in
probability, stochastic analysis and harmonic analysis on groups.
It will also be of interest to mathematicians working in Lie theory
and physicists, statisticians and engineers who are working on
related applications. A background in first year graduate level
measure theoretic probability and functional analysis is essential;
a background in Lie groups and representation theory is certainly
helpful but the first two chapters also offer orientation in these
subjects."
The series is devoted to the publication of monographs and
high-level textbooks in mathematics, mathematical methods and their
applications. Apart from covering important areas of current
interest, a major aim is to make topics of an interdisciplinary
nature accessible to the non-specialist. The works in this series
are addressed to advanced students and researchers in mathematics
and theoretical physics. In addition, it can serve as a guide for
lectures and seminars on a graduate level. The series de Gruyter
Studies in Mathematics was founded ca. 35 years ago by the late
Professor Heinz Bauer and Professor Peter Gabriel with the aim to
establish a series of monographs and textbooks of high standard,
written by scholars with an international reputation presenting
current fields of research in pure and applied mathematics. While
the editorial board of the Studies has changed with the years, the
aspirations of the Studies are unchanged. In times of rapid growth
of mathematical knowledge carefully written monographs and
textbooks written by experts are needed more than ever, not least
to pave the way for the next generation of mathematicians. In this
sense the editorial board and the publisher of the Studies are
devoted to continue the Studies as a service to the mathematical
community. Please submit any book proposals to Niels Jacob.
The latest in this series of Oberwolfach conferences focussed on
the interplay between structural probability theory and various
other areas of pure and applied mathematics such as Tauberian
theory, infinite-dimensional rotation groups, central limit
theorems, harmonizable processes, and spherical data. Thus it was
attended by mathematicians whose research interests range from
number theory to quantum physics in conjunction with structural
properties of probabilistic phenomena. This volume contains 5
survey articles submitted on special invitation and 25 original
research papers.
Heinz Bauer (1928-2002) was one of the prominent figures in Convex
Analysis and Potential Theory in the second half of the 20th
century. The Bauer minimum principle and Bauer's work on Silov's
boundary and the Dirichlet problem are milestones in convex
analysis. Axiomatic potential theory owes him what is known by now
as Bauer harmonic spaces. These Selecta collect more than twenty of
Bauer's research papers including his seminal papers in Convex
Analysis and Potential Theory. Above his research contributions
Bauer is best known for his art of writing survey articles. Five of
his surveys on different topics are reprinted in this volume. Among
them is the well-known article Approximation and Abstract Boundary,
for which he was awarded with the Chauvenet Price by the American
Mathematical Association in 1980.
The book is conceived as a text accompanying the traditional
graduate courses on probability theory. An important feature of
this enlarged version is the emphasis on algebraic-topological
aspects leading to a wider and deeper understanding of basic
theorems such as those on the structure of continuous convolution
semigroups and the corresponding processes with independent
increments. Fourier transformation - the method applied within the
settings of Banach spaces, locally compact Abelian groups and
commutative hypergroups - is given an in-depth discussion. This
powerful analytic tool along with the relevant facts of harmonic
analysis make it possible to study certain properties of stochastic
processes in dependence of the algebraic-topological structure of
their state spaces. In extension of the first edition, the new
edition contains chapters on the probability theory of generalized
convolution structures such as polynomial and Sturm-Liouville
hypergroups, and on the central limit problem for groups such as
tori, p-adic groups and solenoids.
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