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The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields. The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, including mathematical analysis, mathematical physics, algebra. ... Read more

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This book is devoted to the asymptotic theory of differentialequations. Asymptotic theory is an independent and important branch ofmathematical analysis that began to develop at the end of the 19thcentury. Asymptotic methods' use of several important phenomena ofnature can be explained.
The main problems considered in the text are based on the notion of anasymptotic space, which was introduced by the author in his works.Asymptotic spaces for asymptotic theory play analogous roles as metricspaces for functional analysis. It allows one to consider many(seemingly) miscellaneous asymptotic problems by means of the samemethods and in a compact general form.
The book contains the theoretical material and general methods of itsapplication to many partial problems, as well as several new resultsof asymptotic behavior of functions, integrals, and solutions ofdifferential and difference equations.
Audience: The material will be of interest to mathematicians,researchers, and graduate students in the fields of ordinarydifferential equations, finite differences and functional equations,operator theory, and functional analysis. ... Read more

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Fourier analysis has been the inspiration for a technological wave of advances in fields such as imaging processing, financial modeling, algorithms and sequence design. This unified, self-contained book examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the [discrete] Fourier transform to finite Abelian groups.

With countless examples and unique exercise sets at the end of each section, Fourier Analysis on Finite Abelian Groups is a perfect companion to a first course in Fourier analysis. The first chapter provides the fundamental material that is a strong foundation for all subsequent chapters.

Special topics including:

* Computing Eigenvalues of the Fourier transform

* Applications to Banach algebras

* Tensor decompositions of the Fourier transform

* Quadratic Gaussian sums.

This book introduces mathematics students to subjects that are within their reach, but it also has powerful applications that may appeal to advanced researchers and mathematicians. The only prerequisites necessary are group theory, linear algebra, and complex analysis.

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This book deals with linear functional differential equationsand operator theory methods for their investigation.
The main topics are: the equivalence of the input-output stability ofthe equation Lx = &mathsf; and the invertibility of theoperator L in the class of casual operators; the equivalence ofinput-output and exponential stability; the equivalence of thedichotomy of solutions for the homogeneous equation Lx =0 and the invertibility of the operator L; the properties ofGreen's function; the independence of the stability of an equationfrom the norm on the space of solutions; shift invariant functionaldifferential equations in Banach space; the possibility of thereduction of an equation of neutral type to an equation of retardedtype; special full subalgebras of integral and difference operators,and operators with unbounded memory; and the analogue of Fredholm'salternative for operators with almost periodic coefficients whereone-sided invertibility implies two-sided invertibility.
Audience: This monograph will be of interest to students andresearchers working in functional differential equations and operatortheory and is recommended for graduate level courses. ... Read more

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This book shows the deep interaction between two importanttheories: Fredholm and local spectral theory. A particular emphasis isplaced on the applications to multipliers and in particular toconvolution operators. The book also presents some important progress,made in recent years, in the study of perturbation theory for classesof operators which occur in Fredholm theory. ... Read more

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Generalising classical concepts of probability theory, theinvestigation of operator (semi)-stable laws as possible limitdistributions of operator-normalized sums of i.i.d. random variable onfinite-dimensional vector space started in 1969. Currently, thistheory is still in progress and promises interesting applications.Parallel to this, similar stability concepts for probabilities ongroups were developed during recent decades. It turns out that theexistence of suitable limit distributions has a strong impact on thestructure of both the normalizing automorphisms and the underlyinggroup. Indeed, investigations in limit laws led to contractablegroups and - at least within the class of connected groups- to homogeneous groups, in particular to groups that aretopologically isomorphic to a vector space. Moreover, it has beenshown that (semi)-stable measures on groups have a vector spacecounterpart and vice versa.
The purpose of this book is to describe the structure of limit lawsand the limit behaviour of normalized i.i.d. random variables ongroups and on finite-dimensional vector spaces from a common point ofview. This will also shed a new light on the classical situation.Chapter 1 provides an introduction to stability problems on vectorspaces. Chapter II is concerned with parallel investigations forhomogeneous groups and in Chapter III the situation beyond homogeneousLie groups is treated. Throughout, emphasis is laid on the descriptionof features common to the group- and vector space situation.
Chapter I can be understood by graduate students with some backgroundknowledge in infinite divisibility. Readers of Chapters II and III areassumed to be familiar with basic techniques from probability theoryon locally compact groups. ... Read more

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The theory of Kac lagebras and their duality, elaboratedindependently in the seventies by Kac and Vainermann and bythe authors of this book, has nowreached a state ofmaturity which justifies the publication of acomprehensiveand authoritative account in bookform. Further, the topic of"quantum groups" has recently become very fashionable andattractedthe attention of more and more mathematicians andtheoretical physicists. However a good characterization ofquantum groups among Hopf algebras inanalogy to thecharacterization of Lie groups among locally compact groupsis still missing. It is thus very valuable to develop thegeneraltheory as does this book, with emphasis on theanalytical aspects of thesubject instead of the purelyalgebraic ones.While in the Pontrjagin duality theory of locally compactabelian groups a perfect symmetry existsbetween a group andits dual, this is no longer true in the various dualitytheorems of Tannaka, Krein, Stinespring and others dealingwith non-abelian locally compact groups. Kac (1961) andTakesaki (1972)formulated the objective of finding a goodcategory of Hopf algebras, containing the category oflocally compact groups and fulfilling a perfectduality.The category of Kac algebras developed in this bookfullyanswers the original duality problem, while not yetsufficientlynon-unimodular to include quantum groups.This self-contained account of thetheory will be ofinterest to all researchers working in quantumgroups,particularly those interested in the approach by Lie groupsand Lie algebras or by non-commutative geometry, and moregenerally also to those working in C* algebras ortheoretical physics. ... Read more

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The theory of functional equations has been developed in arapid and productive way in the second half of the Twentieth Century.This is due to the fact that the mathematical applications increasedthe number of investigations of newer and newer types of functionalequations. At the same time, the self-development of this theory wasalso very fruitful. The material of this volume reflects very well the complexity andapplicability of the most active research fields. The results andmethods contained give a representative crossection of what isrecently happening in the theory of functional equations. Audience: Researchers and advanced students. ... Read more

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This monograph proposes a new classification of periodicfunctions, based on the concept of generalized derivative, defined byintroducing multiplicators and shifts of the argument into the Fourierseries of the original function. This approach permits theclassification of a wide range of functions, including those of whichthe Fourier series may diverge in integral metric, smooth functions,and infinitely differentiable functions, including analytical andentire ones. These newly introduced classes are then investigatedusing the traditional problems of the theory of approximation. Theresults thus obtained offer a new way to look at classical statementsfor the approximation of differentiable functions, and suggestpossibilities to discover new effects. Audience: valuable reading for experts in the field ofmathematical analysis and researchers and graduate students interestedin the applications of the theory of approximation and Fourier series. ... Read more

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The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. It focuses on integral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes, etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. We provide a list of problems which were open at the time of completion of the book.
Audience: The book is aimed at a rather wide audience, ranging from researchers in functional and harmonic analysis to experts in applied mathematics and prospective students. ... Read more

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Ever since its introduction around 1960 by Kirillov, the orbit method has played a major role in representation theory of Lie groups and Lie algebras. This book contains the proceedings of a conference held from August 29 to September 2, 1988, at the University of Copenhagen, about "the orbit method in representation theory." It contains ten articles, most of which are original research papers, by well-known mathematicians in the field, and it reflects the fact that the orbit method plays an important role in the representation theory of semisimple Lie groups, solvable Lie groups, and even more general Lie groups, and also in the theory of enveloping algebras.

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This volume is devoted to generalizations of the classicalBirkhoff and von Neuman ergodic theorems to semigroup representationsin Banach spaces, semigroup actions in measure spaces, homogeneousrandom fields and random measures on homogeneous spaces. Theergodicity, mixing and quasimixing of semigroup actions andhomogeneous random fields are considered as well. In particularhomogeneous spaces, on which all homogeneous random fields arequasimixing are introduced and studied (the n-dimensionalEuclidean and Lobachevsky spaces with n>=2, and all simple Liegroups with finite centre are examples of such spaces. Also dealt withare applications of general ergodic theorems for the construction ofspecific informational and thermodynamical characteristics ofhomogeneous random fields on amenable groups and for proving generalversions of the McMillan, Breiman and Lee-Yang theorems. A variationalprinciple which characterizes the Gibbsian homogeneous random fieldsin terms of the specific free energy is also proved. The book haseight chapters, a number of appendices and a substantial list ofreferences.
For researchers whose works involves probability theory, ergodictheory, harmonic analysis, measure theory and statistical Physics.
... Read more

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This monograph is devoted to the study of Köthe-Bochner function spaces, an area of research at the intersection of Banach space theory, harmonic analysis, probability, and operator theory. A number of significant results---many scattered throughout the literature---are distilled and presented here, giving readers a comprehensive view of Köthe-Bochner function spaces from the subject's origins in functional analysis to its connections to other disciplines. Key features and topics:* Considerable background material provided, including a compilation of important theorems and concepts in classical functional analysis, as well as a discussion of the Dunford-Pettis Property, tensor products of Banach spaces, relevant geometry, and the basic theory of conditional expectations and martingales * Rigorous treatment of Köthe-Bochner spaces, encompassing convexity, measurability, stability properties, Dunford-Pettis operators, and Talagrand spaces, with a particular emphasis on open problems * Detailed examination of Talagrand's Theorem, Bourgain's Theorem, and the Diaz-Kalton Theorem, the latter extended to arbitrary measure spaces. ... Read more

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This book provides a comprehensive theory of almost periodictype functions with a large number of the applications to differentialequations, functional equations and evolution equations. In addition,it also presents a basic theory on ergodicity and its applications inthe theory of function spectrum, semi group of bounded linearoperators and dynamical systems. It reflects new establishment ofrecent years in the field. This monograph is self-contained, the onlyprerequisite being a basic knowledge of functional analysis andordinary differential equations. It is written for the mathematicianswho wish to learn about the subject and is of interest to thespecialists in the areas of abstract harmonic analysis, functionalanalysis, differential (functional) equations, dynamical system andergodicity. It is also suitable as a textbook for graduates inMathematical Analysis. ... Read more

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This monograph is devoted to the theory of hypercomplex systemswith locally compact basis. Such systems were introduced by Yu.Berezansky and S. Krein in the 1950s and are a generalisation of thenotion of a hypergroup (a family of generalised shift operators) whichwas introduced in the 1970s.
The book gives a state-of-the-art account of hypercomplex systemstheory. After the introductory chapter, it treats the Lie theory ofhypercomplex systems and examples. Topics covered include Fouriertransforms, the Plancherel theorem, the Peter-Weyl theorem,representation theory, duality, Gelfand pairs, Sturm-Liouvilleoperators, and Lie theory. New proofs of results concerningTannaka-Krein duality and Gelfand pairs are given. On the basisof this theory, new approaches to the construction of harmonicanalysis on well-known objects become possible.
Audience: This volume will be of interest to researchers andgraduate students involved in harmonic analysis and representationtheory. ... Read more

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The Pontryagin-van Kampen duality theorem and the Bochner theorem on positive-definite functions are known to be true for certain abelian topological groups that are not locally compact. The book sets out to present in a systematic way the existing material. It is based on the original notion of a nuclear group, which includes LCA groups and nuclear locally convex spaces together with their additive subgroups, quotient groups and products. For (metrizable, complete) nuclear groups one obtains analogues of the Pontryagin duality theorem, of the Bochner theorem and of the Lévy-Steinitz theorem on rearrangement of series (an answer to an old question of S. Ulam). The book is written in the language of functional analysis. The methods used are taken mainly from geometry of numbers, geometry of Banach spaces and topological algebra. The reader is expected only to know the basics of functional analysis and abstract harmonic analysis. ... Read more

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Since its emergence as an important research area in the early 1980s, the topic of wavelets has undergone tremendous development on both theoretical and applied fronts. Myriad research and survey papers and monographs have been published on the subject, documenting different areas of applications such as sound and image processing, denoising, data compression, tomography, and medical imaging. The study of wavelets remains a very active field of research, and many of its central techniques and ideas have evolved into new and promising research areas.

This volume, a collection of invited contributions developed from talks at an international conference on wavelets, is divided into three parts: Part I is devoted to the mathematical theory of wavelets and features several papers on wavelet sets and the construction of wavelet bases in different settings. Part II looks at the use of multiscale harmonic analysis for understanding the geometry of large data sets and extracting information from them. Part III focuses on applications of wavelet theory to the study of several real-world problems.

Overall, the book is an excellent reference for graduate students, researchers, and practitioners in theoretical and applied mathematics, or in engineering.

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This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes.

Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.

... Read more

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This volume will be of great appeal to both advanced students and researchers. For the former, it serves as an effective introduction to three interrelated subjects of analysis: semigroups, Markov processes and elliptic boundary value problems. For the latter, it provides a new method for the analysis of Markov processes, a powerful method clearly capable of extensive further development.