Book Description

A common approach in software engineering is to apply during the design phase a variety of structured techniques like top-down design, decomposition and abstraction, while only subsequently, in the implementation phase, is the design tested to ensure reliability. But this approach neglects that central aspects of software design and program development have a strong formal character which admits tool support for the construction of reliable and correct computer systems based on formal reasoning. This monograph provides much information both for theoreticians interested in algebraic theories, and for software engineers building practically relevant tools. The author presents the theoretical foundations needed for the verification of reactive, sequential infinite-state systems. ... Read more

Book Description

The decomposition of the space L2 (G(Q)\G(A)), where G is a reductive group defined over Q and A is the ring of adeles of Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. This book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step toward understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors also provide essential background in subjects such as: automorphic forms; Eisenstein series; Eisenstein pseudo-series, and their properties.It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, the first written using contemporary terminology. ... Read more

Book Description

Computational Techniques for the Summation of Series is a text on the representation of series in closed form. The book presents a unified treatment of summation of sums and series using function theoretic methods. A technique is developed based on residue theory that is useful for the summation of series of both Hypergeometric and Non-Hypergeometric type. The theory is supported by a large number of examples. The book is both a blending of continuous and discrete mathematics and, in addition to its theoretical base; it also places many of the examples in an applicable setting. This text is excellent as a textbook or reference book for a senior or graduate level course on the subject, as well as a reference for researchers in mathematics, engineering and related fields. ... Read more

Book Description

Wavelets is a carefully organized and edited collection of extended survey papers addressing key topics in the mathematical foundations and applications of wavelet theory. The first part of the book is devoted to the fundamentals of wavelet analysis. The construction of wavelet bases and the fast computation of the wavelet transform in both continuous and discrete settings is covered. The theory of frames, dilation equations, and local Fourier bases are also presented. The second part of the book discusses applications of signal analysis, while the third part covers operator analysis and partial differential equations. Each chapter in these sections provides an up-to-date introduction to such topics as sampling theory, probability and statistics, compression, numerical analysis, turbulence, operator theory, and harmonic analysis.The book is ideal for a general scientific and engineering audience, yet it is mathematically precise. It will be an especially useful reference for harmonic analysts, partial differential equation researchers, signal processing engineers, numerical analysts, fluids researchers, and applied mathematicians. ... Read more

Reviews (1)

A lot of material!
The textbook, actually a collection of lectures, provides a lot of material and examples, and it tries to lead the reader from the basics of subject to the applications of wavelets. Despite of the wide range presented by the chapters, the high reciprocal independence and the completness of proposed path, as an introductory book I would prefer the classical "Ten Lectures on Wavelets", even if Benedetto-Frazier work could be more suitable to get at-a-glance what people are doing (i.e. were doing at the moment of publication) with wavelets. ... Read more

Book Description

Pseudorandom sequences have widespread applications, for instance, in spread spectrum, code division multiple access, optical and ultrawide band communication systems, as well as in ranging systems global positioning systems, circuit testing and stream ciphers. Such sequences also have strong ties to error-correcting codes.
This volume contains survey and research papers on sequences and their applications. It brings together leading experts from discrete mathematics, computer science and communications engineering, and helps to bridge advances in these different areas. Papers in this volume discuss the theory of sequences and their applications in cryptography, coding theory, communications systems, numerical computation and computer simulation.
... Read more

Book Description

This volume contains research and expository papers in a number of different topics including Number Theory, Special Functions, Physics and Combinatorics. A subject that is common to all of these fields is q-series. The chapters contain some of the latest developments in the fields of q-series. Some of the papers describe how symbolic computation has aided these developments. Audience: Researchers and graduate students interested in recent developments in q-series and symbolic computation. ... Read more

Book Description

This book is concerned with the modern theory of Fourier series. Treating developments since Zygmund's classic study, the authors begin with a thorough discussion of the classical one-dimensional theory from a modern perspective. The text then takes up the developments of the 1970s, beginning with Fefferman's famous disc counterexample. The culminating chapter presents Cordoba's geometric theory of Kayeka maximal functions and multipliers. Research workers in the fields of Fourier analysis and harmonic analysis will find this a valuable account of these developments. Second year graduate students, who are familiar with Lebesgue theory and are acquainted with distributions, will be able to use this as a textbook which will bring them up to the exciting open questions in the field. ... Read more

Book Description

This is the first book dedicated to covering the basic elements of the Gibbs phenomenon as it appears in various applications where functions with jump discontinuities are represented. It is presented with detailed analysis and illustrations combined with historical information. The author covers the appearance of the Gibbs phenomenon in Fourier analysis, orthogonal expansions, integral transforms, splines and wavelet approximations. Methods of reducing, or filtering out, such phenomena that cover all the above function representations are also addressed. The book includes a thorough bibliography of some 350 references.Audience: The work is intended as an introduction for engineering and scientific practitioners in the fields where this phenomenon may appear in their use of various function representations. It may also be used by qualified students. ... Read more

Book Description

The Heisenberg group plays an important role in several branches of mathematics, such as representation theory, partial differential equations, number theory, several complex variables, and quantum mechanics. This monograph deals with various aspects of harmonic analysis on the Heisenberg group, which is the most commutative among the non-commutative Lie groups, and hence gives the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The aim of this text is to demonstrate how the standard results of abelian harmonic analysis take shape in the non-abelian setup of the Heisenberg group.

Several results in this monograph appear for the first time in book form, and some theorems have not appeared elsewhere. The detailed discussion of the representation theory of the Heisenberg group goes well beyond the basic Stone-von Neumann theorem, and its relations to classical special functions is invaluable for any reader interested in this group. Topics covered include the Plancherel and Paley-Wiener theorems, spectral theory of the sublaplacian, Wiener-Tauberian theorems, Bochner-Riesz means and multipliers for the Fourier transform.

Thangavelu's exposition is clear and well developed, and leads to several problems worthy of further consideration. Any reader who is interested in pursuing research on the Heisenberg group will find this unique and self-contained text invaluable. ... Read more

Reviews (1)

Great Book!
Excellent choice of topics for Heisenberg group fans! A little advanced but worth the effort. He proves Hardy's Theorem for the Weyl transform ... I found it the most amazing proof and I learnt that it was published in this book for the first time! The material on the sub-Laplacian is treated differently and is very readable. One begins to appreciate the Hermite/Laguerre machinery which may otherwise appear boring. Strongly recommended at the graduate and research level or for fun ;) ... Read more

Book Description

In a series of three papers published in 1930 and 1931, Paley and Zygmund44studied a variety of problems concerning series of independent random functions and raised the question of the uniform convergence a.s. of the random Fourier series. ... Read more

Book Description

The study of wavelets has grown tremendously over the past decade and found myriad applications in areas such as image processing, signal processing, data compression, and numerical methods. This volume contains articles by eminent mathematicians on aspects of wavelet theory. These chapters include a systematic study of the approximation properties of neural networks and a look at several applications where classical approaches were found to be inadequate. INV STAT: Normal ... Read more

Book Description

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

Book Description

This book contains survey papers and research papers by leading experts on sequences and their applications. It discusses both the theory of sequences and their applications in cryptography, coding theory, communications systems, numerical computation and computer simulation. Sequences have important applications in ranging systems, spread spectrum communication systems, multi-terminal system identification, code division multiply access communications systems, global positioning systems, software testing, circuit testing, computer simulation, and stream ciphers. The papers contained in this volume bring together experts from discrete mathematics, computer science and communications engineering, and help to bridge advances in these different areas. ... Read more

Book Description

Harmonic analysis and probability have long enjoyed a mutually beneficial relationship that has been rich and fruitful. This monograph, aimed at researchers and students in these fields, explores several aspects of this relationship. The primary focus of the text is the non-tangential maximal function and the area function of a harmonic function and their probabilistic analogues in martingale theory. The text first gives the requisite background material from harmonic analysis and discusses known results concerning the non-tangential maximal function and area function, as well as the central and essential role these have played in the development of the field. The book next discusses further refinements of traditional results: among these are sharp good-lambda inequalities and laws of the iterated logarithm involving non-tangential maximal functions and area functions. Many applications of these results are given. Throughout, the constant interplay between probabi! lity and harmonic analysis is emphasized and explained. The text contains some new and many recent results combined in a coherent presentation.

Series: Progress in Mathematics, Vol. 175 ... Read more

Book Description

This book contains a detailed exposition of Carleson-Hunt theorem following the proof of Carleson: to this day this is the only one giving better bounds. It points out the motivation of every step in the proof. Thus the Carleson-Hunt theorem becomes accessible to any analyst.The book also contains the first detailed exposition of the fine results of Hunt, Sjölin, Soria, etc on the convergence of Fourier Series. Its final chapters present original material. With both Fefferman's proof and the recent one of Lacey and Thiele in print, it becomes more important than ever to understand and compare these two related proofs with that of Carleson and Hunt. These alternative proofs do not yield all the results of the Carleson-Hunt proof. The intention of this monograph is to make Carleson's proof accessible to a wider audience, and to explain its consequences for the pointwise convergence of Fourier series for functions in spaces near $äcal Lü^1$, filling a well-known gap in the literature. ... Read more