Introduction to the Theory of Distributions

Book Description

The theory of distributions is an extension of classical analysis, an area of particular importance in the field of linear partial differential equations. Underlying it is the theory of topological vector spaces, but it is possible to give a systematic presentation without a knowledge of this. The material in this book, based on graduate lectures given over a number of years requires few prerequisites but the treatment is rigorous throughout. From the outset, the theory is developed in several variables. It is taken as far as such important topics as Schwartz kernels, the Paley-Wiener-Schwartz theorem and Sobolev spaces. In this second edition, the notion of the wavefront set of a distribution is introduced. It allows many operations on distributions to be extended to larger classes and gives much more precise understanding of the nature of the singularities of a distribution. This is done in an elementary fashion without using any involved theories. This account will be useful to graduate students and research workers who are interested in the applications of analysis in mathematics and mathematical physics.

Customer Reviews:

An Excellent Introduction.......2000-11-02

Every physicist and mathematician uses distributions (sometimes called generalized functions), albeit often unknowingly. From its origins in the Dirac delta `function', distribution theory continues to influence many research areas from quantum mechanics to partial differential equations, but has also grown into an important field in its own right. For anyone interested in learning about the field, this is clearly the first port of call. It presents a balanced introduction to the subject on a level suitable for anyone with a basic grounding in analysis (no knowledge of functional analysis is required).

Book Description

This book presents most of the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics. Both the standard C8 pseudodifferential calculus and the analytic microlocal analysis is developed, in a context which remains intentionally global so that only the relevant difficulties of the theory are encountered. The originality lies in the fact that the main features of analytic microlocal analysis are derived from a single and elementary a priori estimate. Various exercises illustrate the chief results of each chapter while introducing the reader to further developments of the theory. This book is aimed at non-specialists of the subject and the only required prerequisite is a basic knowledge of the theory of distributions.

Functional Analysis for Probability and Stochastic Processes: An Introduction

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Delicate Balance between the Two Subjects

Functional Analysis for Probability and Stochastic Processes: An Introduction
Adam Bobrowski
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521831660

Book Description

Designed for students of probability and stochastic processes, as well as for students of functional analysis, specifically, this volume presents some chosen parts of functional analysis that can help clarify probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.

Customer Reviews:

Delicate Balance between the Two Subjects.......2005-10-14

This book was developed for a class taught at Rice University that covered both Functional Analysis and Probablity/Statistics. It required the development of this book that gives a good introduction to functional analysis for statisticians and an introduction to statistics for functional analysts.

The book is positioned so that it can be used for both groups of students. This is a delicate balance that can teach introductions (graduate student level) to both fields without leaving the other half of the class bored. As such it is not a text on either subject, but is inbetween. The sections on statistics are written from a viewpoint familiar to the functional analyst, not from that of a mathematician. And the reverse is true in the sections on functional analysis. This is to complete the view of the two groups typically taking the class.

This is a graduate level course, it provides relatively little background information to the basic mathematics needed.

Introduction to Functional Differential Equations (Applied Mathematical Sciences)

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Deprecated and old

Introduction to Functional Differential Equations (Applied Mathematical Sciences)
Jack K. Hale , and Sjoerd M. Verduyn Lunel
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Ordinary Differential Equations

ASIN: 0387940766

Book Description

The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.

Customer Reviews:

Deprecated and old.......2005-12-04

This is a satisfactory treatment of the theory of functional differential equations. Many of the chapters are not written with students in mind so I could only partially recommend this work to professors of mathematics.

Introduction to Subfactors (London Mathematical Society Lecture Note Series)

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Introduction to Subfactors (London Mathematical Society Lecture Note Series)
V. Jones , and V. S. Sunder
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0521584205

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Subfactors, a subject of considerable research activity for about fifteen years, are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. The authors present a new pictorial approach to subfactors in addition to discussions of basic principles. This book provides an important introduction to the subject and is aimed at students with only a little familiarity with the theory of Hilbert space and other newcomers to the field.

Introduction to Banach Algebras, Operators, and Harmonic Analysis (London Mathematical Society Student Texts)

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Introduction to Banach Algebras, Operators, and Harmonic Analysis (London Mathematical Society Student Texts)
H. Garth Dales , Pietro Aiena , Jörg Eschmeier , Kjeld Laursen , and George A. Willis
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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Book Description

Based on lectures given at an instructional course, this volume enables readers with a basic knowledge of functional analysis to access key research in the field. The authors survey several areas of current interest, making this volume ideal preparatory reading for students embarking on graduate work as well as for mathematicians working in related areas.

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This work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading researchers, give introductions to their subjects at a level ideal for beginning graduate students, and others interested in the subject. The collection has been carefully edited so as to form a coherent and accessible introduction to current research topics. The first chapter by Professor Dales introduces the general theory of Banach algebras, which serves as a background to the remaining material. Dr Willis then studies a centrally important Banach algebra, the group algebra of a locally compact group. The remaining chapters are devoted to Banach algebras of operators on Banach spaces: Professor Eschmeier gives all the background for the exciting topic of invariant subspaces of operators, and discusses some key open problems; Dr Laursen and Professor Aiena discuss local spectral theory for operators, leading into Fredholm theory.

An Introduction To K-theory For C (London Mathematical Society Student Texts)

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Nice place to start...

An Introduction To K-theory For C (London Mathematical Society Student Texts)
F. Larsen
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521789443

Book Description

Over the past twenty-five years K-theory has become an integrated part of the study of C*-algebras. This book gives a very elementary introduction to this interesting and rapidly growing area of mathematics. The authors cover the basic properties of the functors K and K1 and their interrelationship. In particular, the Bott periodicity theorem is proved (Atiyah's proof), and the six-term exact sequence is derived. The theory is well illustrated with 120 exercises and examples, making the book ideal for beginning graduate students in functional analysis, especially operator algebras, and for researchers from other areas of mathematics who want to learn about this subject.

Customer Reviews:

Nice place to start..........2003-11-07

It's a very clear book with virtually no typos/mistakes and lots of nice exercises (if you are willing to do them - highly recommended). Really an introductory text, so you might need to consult the more advanced books of Wegge-Olsen, Blackadar after this one for more information on the subject. Nice place to start, especially if it is for self-study.

An Introduction to Infinite Dimensional Systems (UNIVERSITEXT)

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An Introduction to Infinite Dimensional Systems (UNIVERSITEXT)
Prato Da.G
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 3540290206

Book Description

In this revised and extended version of his course notes from a 1-year course at Scuola Normale Superiore, Pisa, the author provides an introduction â€“ for an audience knowing basic functional analysis and measure theory but not necessarily probability theory â€“ to analysis in a separable Hilbert space of infinite dimension.

Starting from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate some basic stochastic dynamical systems (including dissipative nonlinearities) and Markov semi-groups, paying special attention to their long-time behavior: ergodicity, invariant measure. Here fundamental results like the theorems of Prokhorov, Von Neumann, Krylov-Bogoliubov and Khas'minski are proved. The last chapter is devoted to gradient systems and their asymptotic behavior.

Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE's and the Theory of Global Attractors (CAMBRIDGE TEXTS IN APPLIED MATHEMATICS)

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Masters Project

Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE's and the Theory of Global Attractors (CAMBRIDGE TEXTS IN APPLIED MATHEMATICS)
James C. Robinson
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0521635640

Book Description

This book develops the theory of global attractors for a class of parabolic PDEs that includes reaction-diffusion equations and the Navier-Stokes equations, two examples that are treated in detail. A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment of existence and uniqueness of solutions for both linear time-independent problems (Poisson's equation) and the nonlinear evolution equations which generate the infinite-dimensional dynamical systemss of the title. Attention then switches to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space which determines the asymptotic dynamics. In particular, the concluding chapters investigate in what sense the dynamics restricted to the attractor are themselves "finite-dimensional." The book is intended as a didactic text for first year graduates, and assumes only a basic knowledge of Banach and Hilbert spaces, and a working understanding of the Lebesgue integral.

Customer Reviews:

Masters Project.......2007-01-11

I am using this book in filling in the gaps in my knowledge of PDEs for my masters project and it is great. I have to agree with this author that this is a great book for a beginning student in the field. The way the auther builds up the material of functional analysis for the study of PDEs really cleared up a lot for me.

Introduction to Hp Spaces (Cambridge Tracts in Mathematics)

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Introduction to Hp Spaces (Cambridge Tracts in Mathematics)
Paul Koosis
Manufacturer: Cambridge University Press
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Binding: Hardcover

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ASIN: 0521455219

Book Description

The first edition of this well-known book was noted for its clear and accessible exposition of the basic theory of Hardy spaces from the concrete point of view (in the unit circle and the half plane). This second edition retains many of the features found in the first--detailed computation, an emphasis on methods--but greatly extends its coverage. The discussions of conformal mapping now include Lindelöf's second theorem and the one due to Kellogg. A simple derivation of the atomic decomposition for RH1 is given, and then used to provide an alternative proof of Fefferman's duality theorem. Two appendices by V.P. Havin have also been added: on Peter Jones' interpolation formula for RH1 and on Havin's own proof of the weak sequential completeness of L1/H1(0). Numerous other additions, emendations and corrections have been made throughout.

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