Birkhäuser Advanced Texts

Transcription

1

2 Birkhäuser Advanced Texts Basler Lehrbücher Edited by Herbert Amann, University of Zürich Steven G. Krantz, Washington University, St. Louis Shrawan Kumar, University of North Carolina at Chapel Hili

3 Steven G. Krantz Harold R. Parks APrimerof Real Analytic Functions Second Edition Springer Science+Business Media, LLC

4 Steven G. Krantz Washington University Department of Mathematics St. Louis, MO U.S.A. Harold R. Parks Oregon State University Department of Mathematics Corvallis, OR U.S.A. Library of Congress CataIoging-in-Pubücation Data A CIP catalogue record for Ibis book is available from the Library of Congress, Washington D.C., USA. AMS Subject Classifications: Primary: 26E05, 30BIO, 32C05; Secondary: 14PI5, 26A99. 26BIO, 26B40, 26EIO, 30B40, 32C09, 35AIO, 54C30 Printed on acid-free paper 1t>2002 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 2002 Softcover reprint ofthe hardcover 2nd edition 2002 Birkhäuser All rights reserved. This work may not be translated or copied in whole or in part without the wriuen permission of the publisher (Springer Science+Business Media, LLC). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such. is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. ISBN ISBN (ebook) SPIN DOI / Reformatted from the author's files by TEXniques, Inc., Carnbridge. MA I

5 To the memory of Frederick J. Almgren, Jr. ( ), teacher and friend

6 Contents Preface to the Second Edition ix Preface to the First Edition 1 Elementary Properties 1.1 Basic Properties of Power Series 1.2 Analytic Continuation The Formula of Faa di Bruno Composition of Real Analytic Functions 1.5 Inverse Functions xi Multivariable Calculus of Real Analytic Functions 2.1 Power Series in Several Variables Real Analytic Functions of Several Variables. 2.3 The Implicit Function Theorem A Special Case of the Cauchy-Kowalewsky Theorem 2.5 The Inverse Function Theorem Topologies on the Space of Real Analytic Functions. 2.7 Real Analytic Submanifolds Bundles over areal Analytic Submanifold 2.8 The General Cauchy-Kowalewsky Theorem

7 viii Contents 3 Classical Topics Introductory Remarks The Theorem ofpringsheim and Boas Besicovitch's Theorem Whitney's Extension and Approximation Theorems The Theorem of S. Bernstein Some Questions of Hard Analysis Quasi-analytic and Gevrey Classes Puiseux Series Separate Real Analyticity Results Motivated by Partial Differential Equations Division ofdistributions I Projection of Polynomially Defined Sets Division of Distributions II The FBI Transform The Paley-Wiener Theorem Topics in Geometry The Weierstrass Preparation Theorem Resolution of Singularities Lojasiewicz's Structure Theorem for Real Analytic Varieties The Embedding of Real Analytic Manifolds Semianalytic and Subanalytic Sets Basic Definitions Facts Conceming Semianalytic and Subanalytic Sets Examples and Discussion Rectilinearization Bibliography 187 Index 203

8 Preface to the Second Edition It is a pleasure and a privilege to write this new edition of A Primer 0/ Real Analytic Functions. The theory of real analytic functions is the wellspring of mathematical analysis. It is remarkable that this is the first book on the subject, and we want to keep it up to date and as correct as possible. With these thoughts in mind, we have utilized helpful remarks and criticisms from many readers and have thereby made numerous emendations. We have also added material. There is a now a treatment of the Weierstrass preparation theorem, a new argument to establish Hensel's lemma and Puiseux's theorem, a new treatment of Faa di Bruno's forrnula, a thorough discussion of topologies on spaces of real analytic functions, and a second independent argument for the implicit function theorem. We trust that these new topics will make the book more complete, and hence a more useful reference. It is a pleasure to thank our editor, Ann Kostant of Birkhäuser Boston, for making the publishing process as smooth and trouble-free as possible. We are grateful for useful communications from the readers of our first edition, and we look forward to further constructive feedback. Steven G. Krantz Harold R. Parks May, 2002

9 Preface to the First Edition The subject of real analytic functions is one of the oldest in mathematical analysis. Today it is encountered early in one's mathematical training: the first taste usually comes in calculus. While most working mathematicians use real analytic functions from time to time in their work, the vast lore of real analytic functions remains obscure and buried in the literature. lt is remarkable that the most accessible treatment of Puiseux's theorem is in Lefschetz's quite old Algebraic Geometry, that the clearest discussion of resolution of singularities for real analytic manifolds is in a book review by Michael Atiyah, that there is no comprehensive discussion in print of the embedding problem for real analytic manifolds. We have had occasion in our collaborative research to become acquainted with both the history and the scope of the theory of real analytic functions. lt seems both appropriate and timely for us to gather together this information in a single volume. The material presented here is of three kinds. The elementary topics, covered in Chapter 1, are presented in great detail. Even results like areal analytic inverse function theorem are difficult to find in the literature, and we take pains here to present such topics carefully. Topics of middling difficulty, such as separate real analyticity, Puiseux series, the FBI transform, and related ideas (Chapters 2-4), are covered thoroughly but rather more briskly. Finally there are some truly deep and difficult topics: embedding of real analytic manifolds, suband semi-analytic sets, the structure theorem fot real analytic varieties, and resolution of singularities are discussed and described. But thorough proofs in these areas could not possibly be provided in a volume of modest length.

10 xii Preface to the First Edition Our aim, therefore, has been to provide an introduction to and a map (a primer if you will) of the subject of real analytic functions. Perhaps this monograph will help to bring to light a diverse and important literature. It is a pleasure to thank Richard Beals, Edward Bierstone, Brian Blank, Harold Boas, Ralph Boas, Josef Siciak, Kennan T. Smith, David Tartakoff, and Michael E. Taylor for many useful comments and suggestions. Of course the responsibility for all remaining errors remains the province of the authors. Steven G. Krantz Harold R. Parks 1992

11 APrimerof Real Analytic Functions Second Edition