Birkhauser Advanced Texts Basler Lehrbticher

Edited by

Herbert Amann, University of ZUrich

Shrawan Kumar, University of North Carolina at Chapel Hill

Ricardo Estrada Ram P. Kanwal

A Distributional Approach to Asymptotics Theory and Applications

Second Edition

Springer Science+Business Media, LLC

Ricardo Estrada Ram P. Kanwal Escuela de Matematica Universidad de Costa Rica San Jose, Costa Rica

Department of Mathematics Pennsylvania State University University Park, PA 16802 USA

Library of Congress Cataloging-in-Publication Data

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA.

AMS Subject Classifications: 30E15, 34Exx, 34E05, 34EIO, 40Cxx, 41A60, 42Axx, 46Fxx, 62E20

Printed on acid-free paper. © 2002 Springer Science+Business Media New York Originally Published by Birkhăuser Boston in 2002 Softcover reprint of the hardcover 2nd edition 2002

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ISBN 978-1-4612-6410-1 ISBN 978-0-8176-8130-2 (eBook) SPIN 10733825 DOI 10.1007/978-0-8176-8130-2

Formatted from authors' files by TEXniques, Inc., Cambridge, MA.

987 6 5 432 1

To Ana and to my children, Tomas, Rolando, and Ana

-Ricardo Estrada

To my grandchildren, Allison, Mallory and Trevor

and to my friend, Jagdish

-Ram P. Kanwal

Contents

Preface xi

1 Basic Results in Asymptotics 1 1.1 Introduction.... 1 1.2 Order Symbols ..... 2 1.3 Asymptotic Series . . . . 8 1.4 Algebraic and Analytic Operations 14 1.5 Existence of Functions with a Given Asymptotic Expansion . 18 1.6 Asymptotic Power Series in a Complex Variable. 21 1.7 Asymptotic Approximation of Partial Sums 26 1.8 The Euler-Maclaurin Summation Formula . 36 1.9 Exercises ................ 43

2 Introduction to the Theory of Distributions S3 2.1 Introduction............ 53 2.2 The Space of Distributions V' ..... 55 2.3 Algebraic and Analytic Operations . . . 59 2.4 Regularization, Pseudofunction and Hadamard Finite Part . 63 2.5 Support and Order. . . . . . . . . . . . . . . . . . . . 70 2.6 Homogeneous Distributions .............. 72 2.7 Distributional Derivatives of Discontinuous Functions. 76 2.8 Tempered Distributions and the Fourier Transform. . . 82 2.9 Distributions of Rapid Decay . . . . . . . . . . . . . . 87 2.10 Spaces of Distributions Associated with an Asymptotic Sequence. 90 2.11 Exercises ............. . . . . . . . . . . . . .. 94

viii Contents

3 A Distributional Theory for Asymptotic Expansions 107 107 108 113 121 126 132 142 148 152 161 169 170 172

3.1 Introduction ............. . 3.2 The Taylor Expansion of Distributions 3.3 The Moment Asymptotic Expansion 3.4 Expansions in the Space pi ... 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 (Ax) as A ~ 00 in Other Cases 3.10 Asymptotic Separation of Variables ..

3.10.1 Local Behavior ....... . 3.10.2 Quasiasymptotic Expansions.

3.11 Exercises . . . . . . . . . . . . . . .

4 Asymptotic Expansion of Multidimensional Generalized Functions 187 4.1 Introduction....................... 187 4.2 Taylor Expansion in Several Variables . . . . . . . . . 188 4.3 The Multidimensional Moment Asymptotic Expansion 191 4.4 Laplace's Asymptotic Formula 197 4.5 Fourier Type Integrals. . . . . 205 4.6 Time-DomainAsymptotics.. 213

4.6.1 Interior Critical Points 214 4.6.2 Boundary Critical Points 216 4.6.3 Corners......... 218

4.7 Further Examples . . . . . . . . 219 4.8 Tensor Products and Partial Asymptotic Expansions 222 4.9 An Application in Quantum Mechanics. . . 227 4.10 Expansion of Kernels of the Type f (Ax, x) 232

4.10.1 The Operator tl . . . . . . . . . . . . 233 4.10.2 The Moment Expansion of f (Ax, x) . 234 4.10.3 The Expansion in the Other Cases 237

4.11 Exercises ................... 242

5 Asymptotic Expansion of Certain Series Considered by Ramanujan 249 5.1 Introduction..... 249 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 ...... .

250 255 264 273 284 292

Contents

6 Cesaro Behavior of Distributions 6.1 Introduction......... 6.2 Summability of Series and Integrals

6.2.1 Abel summabiIity . . . . . . 6.2.2 HOlder-Cesaro-Riesz Summability .

6.3 The Behavior of Distributions in the (C) Sense 6.4 The Cesaro Summability of Evaluations . . 6.5 Parametric Behavior ........... . 6.6 Characterization of Tempered Distributions 6.7 The Space 1(' . . . . . . . .

6.8 Spherical Means. . . . . . . . . . . . . . . 6.9 Existence of Regularizations . . . . . . . .

6.9.1 Regularization in Several Dimensions 6.9.2 Singular Hypersurfaces .

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 .

6.15.1 Asymptotically Homogeneous Functions 6.15.2 Primitives of Null Sequences .. 6.15.3 Characterization of Point Values

6.16 Spectral Asymptotics ....... . 6.16.1 A Special Case ...... .

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 .

ix

297 297 298 299 301 305 308 312 318 319 323 325 330 332 334 340 346 352 357 361 362 364 365 368 373 375 382 386 390

397 397 398 399

7.4 Dual Taylor Series as Asymptotics of Solutions of Equations 409 7.5 Boundary Layers . . . . . . . . . . 410

7.5.1 Initial value problems '" 411 7.5.2 Inner-Outer Decomposition 414 7.5.3 Boundary Value Problems 418

7.6 Spectral Content Asymptotics . 422 7.7 Exercises . . . . . . . . . . . . . 430

References 433

Index 447

Preface

This book is a revised and expanded version of our book Asymptotic Analysis: a Distributional Approach [88]. The present work contains many more examples and covers many more topics, comprising numerous results that have appeared in the literature in recent years. We have included several new sections and a whole new chapter on the Cesaro behavior of distributions, a concept that has been found to playa key role in the asymptotic expansion of generalized functions. Lastly, we have incorporated numerous exercises at the end of every chapter; some of the exercises enhance the text material, others give applications, while others, when solved, become almost new sections of this beautiful subject.

Asymptotic analysis is an old subject which has found applications in various fields of pure and applied mathematics, physics, and engineering. For instance, asymptotic techniques are used to approximate very complicated integral expres­sions that result from the transform analysis. Similarly, the solutions of differential equations can often be computed with great accuracy by taking the sum of a few terms of the divergent series obtained by asymptotic calculus. In view of the im­portance of these methods, many excellent books on this subject are available [37], [39], [53], [128], [158], [161], [181], [202].

An important feature of the theory of asymptotic expansions is that experience and intuition play an important part because particular problems are rather individual in nature. Our aim is to present a systematic and simplified approach to this theory by the useof distributions (generalized functions). The theory of distributions is another important area of applied mathematics, which has also found many applications in mathematics, physics and engineering. It is only recently, however, that the close ties between asymptotic analysis and the theory of distributions have been studied in detail [28], [84], [85], [151], [165], [200], [211]. As it turns out, generalized

xii Preface

functions provide a very appropriate framework for asymptotic analysis, where many analytical operations can be performed, and also provide a systematic procedure to assign values to the divergent integrals that often appear in the literature.

This book is suitable for a one semester graduate course in the mathematical and physical sciences. We have offered courses inspired by this material at our respective universities. Although the book is based on our own research, we have attempted to make the material self-contained. With this goal in mind we incorporated one chapter on the introduction of classical asymptotic analysis and one chapter on the basic principles of generalized functions as needed in the sequel.

The material is divided into seven chapters. In Chapter 1 we explain the classi­cal principles of the asymptotic theory. The content of this chapter is designed to motivate and prepare the reader for subsequent chapters. We discuss Landau order symbols and then the asymptotic series, highlighting the fact that they are usually divergent, and explain the operations that can be applied to them. We also explain the existence of functions with a given asymptotic development, the asymptotic power series, the approximation of partial sums, and the celebrated Euler-Maclaurin sum­mation formula.

Since generalized functions permeate our presentation, we explain all the basic principles of this fascinating field in Chapter 2. We have taken great care in explaining all the function spaces that we need in presenting the theory. We have also tried to describe the various algebraic and analytic operations that can be applied to the distributions, including their Fourier transforms. We also consider the distributions of rapid decay and the distributions associated with an asymptotic sequence - topics of great importance in this work. These first two chapters remain quite similar to the corresponding ones of [88] although several new examples are added.

Chapter 3 is devoted to the theme and methodology of our theory. We start with the moment asymptotic expansion of generalized functions. As we show, the moment expansion holds in a wide variety of situations, particularly for distributions of fast decay and for distributions of rapid oscillation. The moment asymptotic expansion immediately yields the asymptotic development of several integrals and series. We illustrate these ideas with many examples, deriving, in particular, results such as Watson's lemma for the expansion of Laplace transforms and Stirling's approxima­tion of n!. The use of the notion of change of variables in distributions allows us to obtain the expansion of more complicated distributional kernels, which in tum provide the most important methods for the asymptotic development of integrals, namely, the Laplace formula, the method of stationary phase and the method of steep­est descent. Then we present the distributional analysis of time-domain asymptotics, an extension of the method of stationary phase obtained recently [33], [90]. We also consider the situation when the moment expansion does not hold, and show that the expansions should be given in terms of homogeneous and associated homogeneous functions.

We then study the asymptotic separation of variables and the concept of local distributional expansion, the notion of distributional point value, in particular. The quasiasymptotic expansions, a different but complementary approach to the study of the relationship between distributions and asymptotics, developed by the Novi Sad

Preface xiii

school [166], is also considered in this chapter. To make the presentation accessible, in Chapter 3 we limit ourselves to the one-dimensional theory.

In Chapter 4 we extend the analysis of the previous chapter to corresponding multidimensional problems. As in Chapter 3, we start with the moment asymptotic expansion, and using the notion of change of variables in distributions, we obtain the Laplace asymptotic formula in several variables. We also use these ideas to derive the asymptotic development of oscillatory integrals, the so-called Fourier type integrals. We added a section on time-domain asymptotics in several variables. Further, we employ the theory of topological tensor products to obtain the asymptotic expansion of vector-valued generalized functions. This helps us in developing the theory of partial asymptotic expansions. Partial expansions are a powerful tool which provide a way to obtain rather sharp multidimensional developments. Moreover, we illustrate the use of partial expansions with an application to quantum mechanical twisted products. We also study whether the expansion of kernels of the form f (Ax, x) can be obtained from the partial expansion of f (Ax, y) by setting x = y [196], a question of great importance in the study of singular operators [26], [27]. Another interesting feature of this chapter is the derivation of the far-field behavior of potential and scattering fields. We find that there arise many nonclassical terms, in addition to previously known classical terms.

In Chapter 5 we present the asymptotic development of certain functions defined by series containing a small parameter, which are of importance in number theory and, as we show throughout the book, in many other contexts as well. Series of this kind were first studied by Ramanujan [170], who gave their asymptotic expansion. However, as in much of Ramanujan's work, these results are given without any proofs. Proofs of some of these results have been provided by appealing to Mellin transforms [16]. By using our methods, we are able to present proofs ofRamamujan's results and of many interesting generalizations for a wide class offunctions. We study not only the one-dimensional series but the multidimensional series as well. Among many other concepts we introduce the notion of a distributionally small sequence and relate it to the classical principles of summability; these ideas are extended in Chapter 6 to general distributions. We added a section on the asymptotic expansion of the standard partition function, showing how the celebrated Hardy-Ramanujan asymptotic approximation follows in a simple way from our techniques.

Chapter 6 is new. It is concerned with the topic that has proved to be pivotal in the study of asymptotic expansions of generalized functions, namely, the Cesaro behavior of distributions [69]. Indeed, the use of summability ideas allows us to identify the distributions that satisfy the moment asymptotic expansion as precisely the distributions of rapid distributional decay at infinity in the Cesaro sense, and to characterize them as the elements of the distribution space J('. The chapter starts with a survey of the classical ideas of Abel and Cesaro summability. Then these notions are extended to the distributional framework. The characterization of spaces of distributions, such as the space J(' already mentioned, and the space of tem­pered distributions S' follow. We later demonstrate how the results generalize to distributions of several variables by employing spherical means.

xiv Preface

The second part of Chapter 6 presents several related ideas, whose study is fa­cilitated by the summability notions. We give a simple necessary and sufficient condition for the existence of distributional regularizations, an extension of the in­tegral test to the summability framework, and the analytic continuation of moment functions and zeta functions. We also discuss the summability of the Fourier series of periodic distributions, starting with some classical results and advancing up to the characterization of the Fourier series of generalized functions having distributional point values [66]. The final sections of this chapter are concerned with spectral asymptotics of operators. We show how our analysis permits one to understand the average type expansions that arise in the eigenvalue distribution and in the small time behavior of Green functions. These expansions, particularly the expansion of the coincidence limits, are important in the study of noncommutative geometry [106].

Chapter 7 surveys the use of divergent series of delta functions in various con­texts. Series of Dirac delta functions appear rather frequently in the previous chapters since they form the basic blocks in the moment asymptotic expansion of generalized functions. Furthermore, they have been used as formal tools in other branches of applied mathematics, such as the solution of differential and functional equations, the construction of weight functions for orthogonal polynomials, and the solution of moment problems. Our aim is to give a rigorous interpretation of these formal methods in the framework of asymptotic analysis. We study the use of series of deltas in the solution of differential equations. We also show how singular pertur­bation problems, especially boundary layer problems, can be studied by the use of such divergent series of generalized functions. We indicate how spectral content asymptotics can be treated by using the boundary layer theory and the series of deltas.

We would like to express our thanks to several colleagues that made sugges­tions on how [88] could be improved, particularly J. BrUning, S. A. Fulling, J. M. Gracia-Bondfa, and J. C. Varilly. We would also like to thank Ann Kostant and Tom Grasso of Birkhauser, and Elizabeth ofT}3Xniques, for their continuing patience and cooperation.

Ricardo Estrada Ram P. Kanwal

A Distributional Approach to Asymptotics

Theory and Applications