A Distributional Approach to Asymptotics: Theory and Applications / Edition 2

by Ricardo Estrada, Ram P. Kanwal

Overview

"...The authors of this remarkable book are among the very few who have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book...[it] should be on your bookshelf if you are at all interested in knowing what an asymptotic series is." -"The Bulletin of Mathematics Books" (Review of the 1st edition) ** "...The book is a valuable one, one that many applied mathematicians may want to buy. The authors are undeniably experts in their field...most of the material has appeared in no other book." -"SIAM News" (Review of the 1st edition) This book is a modern introduction to asymptotic analysis intended not only for mathematicians, but for physicists, engineers, and graduate students as well. Written by two of the leading experts in the field, the text provides readers with a firm grasp of mathematical theory, and at the same time demonstrates applications in areas such as differential equations, quantum mechanics, noncommutative geometry, and number theory. Key features of this significantly expanded and revised second edition: * addition of a new chapter and many new sections * wide range of topics covered, including the Ces.ro behavior of distributions and their connections to asymptotic analysis, the study of time-domain asymptotics, and the use of series of Dirac delta functions to solve boundary value problems * novel approach detailing the interplay between underlying theories of asymptotic analysis and generalized functions * extensive examples and exercises at the end of each chapter * comprehensive bibliography and index This work is an excellent tool for the classroom and an invaluable self-study resource that will stimulate application of asymptotic

Product Details

ISBN-13:	9780817641429
Publisher:	Birkhäuser Boston
Publication date:	02/08/2002
Series:	Birkhäuser Advanced Texts Basler Lehrbücher
Edition description:	2nd ed. 2002
Pages:	454
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

1 Basic Results in Asymptotics.- 1.1 Introduction.- 1.2 Order Symbols.- 1.3 Asymptotic Series.- 1.4 Algebraic and Analytic Operations.- 1.5 Existence of Functions with a Given Asymptotic Expansion.- 1.6 Asymptotic Power Series in a Complex Variable.- 1.7 Asymptotic Approximation of Partial Sums.- 1.8 The Euler-Maclaurin Summation Formula.- 1.9 Exercises.- 2 Introduction to the Theory of Distributions.- 2.1 Introduction.- 2.2 The Space of Distributions D?.- 2.3 Algebraic and Analytic Operations.- 2.4 Regularization, Pseudofunction and Hadamard Finite Part.- 2.5 Support and Order.- 2.6 Homogeneous Distributions.- 2.7 Distributional Derivatives of Discontinuous Functions.- 2.8 Tempered Distributions and the Fourier Transform.- 2.9 Distributions of Rapid Decay.- 2.10 Spaces of Distributions Associated with an Asymptotic Sequence.- 2.11 Exercises.- 3 A Distributional Theory for Asymptotic Expansions.- 3.1 Introduction.- 3.2 The Taylor Expansion of Distributions.- 3.3 The Moment Asymptotic Expansion.- 3.4 Expansions in the Space P?.- 3.5 Laplace’s Asymptotic Formula.- 3.6 The Method of Steepest Descent.- 3.7 Expansion of Oscillatory Kernels.- 3.8 Time-Domain Asymptotics.- 3.9 The Expansion of f (?x) as——— in Other Cases.- 3.10 Asymptotic Separation of Variables.- 3.11 Exercises.- 4 Asymptotic Expansion of Multidimensional Generalized Functions.- 4.1 Introduction.- 4.2 Taylor Expansion in Several Variables.- 4.3 The Multidimensional Moment Asymptotic Expansion.- 4.4 Laplace’s Asymptotic Formula.- 4.5 Fourier Type Integrals.- 4.6 Time-Domain Asymptotics.- 4.7 Further Examples.- 4.8 Tensor Products and Partial Asymptotic Expansions.- 4.9 An Application in Quantum Mechanics.- 4.10 Expansion of Kernels of the Type f (?x, x).- 4.11 Exercises.- 5 AsymptoticExpansion of Certain Series Considered by Ramanujan.- 5.1 Introduction.- 5.2 Basic Formulas.- 5.3 Lambert Type Series.- 5.4 Distributionally Small Sequences.- 5.5 Multiple Series.- 5.6 Unrestricted Partitions.- 5.7 Exercises.- 6 Cesàro Behavior of Distributions.- 6.1 Introduction.- 6.2 Summability of Series and Integrals.- 6.3 The Behavior of Distributions in the (C) Sense.- 6.4 The Cesàro Summability of Evaluations.- 6.5 Parametric Behavior.- 6.6 Characterization of Tempered Distributions.- 6.7 The Space K?.- 6.8 Spherical Means.- 6.9 Existence of Regularizations.- 6.10 The Integral Test.- 6.11 Moment Functions.- 6.12 The Analytic Continuation of Zeta Functions.- 6.13 Fourier Series.- 6.14 Summability of Trigonometric Series.- 6.15 Distributional Point Values of Fourier Series.- 6.16 Spectral Asymptotics.- 6.17 Pointwise and Average Expansions.- 6.18 Global Expansions.- 6.19 Asymptotics of the Coincidence Limit.- 6.20 Exercises.- 7 Series of Dirac Delta Functions.- 7.1 Introduction.- 7.2 Basic Notions.- 7.3 Several Problems that Lead to Series of Deltas.- 7.4 Dual Taylor Series as Asymptotics of Solutions of Equations.- 7.5 Boundary Layers.- 7.6 Spectral Content Asymptotics.- 7.7 Exercises.- References.

Editorial Reviews

"This is not just a ‘Second Edition’ of some monograph in the usual sense, but a revised and largely expanded version of Asymptotic Analysis: A Distributional Approach (1994) by the same authors…. A completely new chapter on the Cesáro behavior of distributions has been added; moreover there are several new sections, among them respective problem sections at the end of each chapter. Finally, a large number of recent results and additional examples have been included…. Even more than its predecessor, this book presents an interesting and carefully written introduction into the theory and applications of asymptotic analysis based on distribution theory."

—MONATSHEFTE FÜR MATHEMATIK (Review of the Second Edition)

"The authors of this remarkable book are among the very few that have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book...[it] should be on your bookshelf if you are at all interested in knowing what an asymptotic series is."

—THE BULLETIN OF MATHEMATICS BOOKS (Review of the First Edition)

". . . the book is a valuable one, one that many applied mathematicians may want to buy. The authors are undeniably experts in their field . . . most of the material has appeared in no other book."

—SIAM REVIEW (Review of the First Edition)

From the Publisher