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More About This Textbook
Overview
This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject.
Editorial Reviews
Endeavour
Functional Analysis by Elias Stein and Rami Shakarchi is a fastpaced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left.— John D. Cook
Choice
Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off.MathSciNet
This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.— Stevan Pilipovic
Mathematical Reviews
This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semesterlong course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.— Stevan Pilipovit
Endeavour  John D. Cook
Functional Analysis by Elias Stein and Rami Shakarchi is a fastpaced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left.MathSciNet  Stevan Pilipovic
This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.Mathematical Reviews  Stevan Pilipovit
This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semesterlong course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.From the Publisher
"Functional Analysis by Elias Stein and Rami Shakarchi is a fastpaced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left."—John D. Cook, Endeavour blog"Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off."—Choice
"This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."—Stevan Pilipovic, MathSciNet, Mathematical Reviews on the Web
"This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semesterlong course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."—Stevan Pilipovit, Mathematical Reviews
Product Details
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Meet the Author
Elias M. Stein is the Albert Baldwin Dod Professor of Mathematics at Princeton University. Rami Shakarchi received his PhD in mathematics from Princeton University. They are the coauthors of "Complex Analysis, Fourier Analysis", and "Real Analysis" (all Princeton).
Table of Contents
Foreword vii
Preface xvii
Chapter 1 L^{p} Spaces and Banach Spaces 1
1 L^{p} spaces 2
1.1 The Hölder and Minkowski inequalities 3
1.2 Completeness of L^{p} 5
1.3 Further remarks 7
2 The case p = ∞ 7
3 Banach spaces 9
3.1 Examples 9
3.2 Linear functionals and the dual of a Banach space 11
4 The dual space of L^{p} when 1 ≤ p < ∞ 13
5 More about linear functionals 16
5.1 Separation of convex sets 16
5.2 The HahnBanach Theorem 20
5.3 Some consequences 21
5.4 The problem of measure 23
6 Complex L^{p} and Banach spaces 27
7 Appendix: The dual of C(X) 28
7.1 The case of positive linear functionals 29
7.2 The main result 32
7.3 An extension 33
8 Exercises 34
9 Problems 43
Chapter 2 L^{p} Spaces in Harmonic Analysis 47
1 Early Motivations 48
2 The Riesz interpolation theorem 52
2.1 Some examples 57
3 The L^{p} theory of the Hilbert transform 61
3.1 The L^{2} formalism 61
3.2 The L^{p} theorem 64
3.3 Proof of Theorem 3.2 66
4 The maximal function and weaktype estimates 70
4.1 The L^{p} inequality 71
5 The Hardy space H_{r}^{1} 73
5.1 Atomic decomposition of H_{r}^{1} 74
5.2 An alternative definition of H_{r}^{1} 81
5.3 Application to the Hilbert transform 82
6 The space H_{r}^{1} and maximal functions 84
6.1 The space BMO 86
7 Exercises 90
8 Problems 94
Chapter 3 Distributions: Generalized Functions 98
1 Elementary properties 99
1.1 Definitions 100
1.2 Operations on distributions 102
1.3 Supports of distributions 104
1.4 Tempered distributions 105
1.5 Fourier transform 107
1.6 Distributions with point supports 110
2 Important examples of distributions 111
2.1 The Hilbert transform and pv(1/x) 111
2.2 Homogeneous distributions 115
2.3 Fundamental solutions 125
2.4 Fundamental solution to general partial differential equations with constant coefficients 129
2.5 Parametrices and regularity for elliptic equations 131
3 CalderónZygmund distributions and L^{p} estimates 134
3.1 Defining properties 134
3.2 The L^{p} theory 138
4 Exercises 145
5 Problems 153
Chapter 4 Applications of the Baire Category Theorem 157
1 The Baire category theorem 158
1.1 Continuity of the limit of a sequence of continuous functions 160
1.2 Continuous functions that are nowhere differentiable 163
2 The uniform boundedness principle 166
2.1 Divergence of Fourier series 167
3 The open mapping theorem 170
3.1 Decay of Fourier coefficients of L^{1}functions 173
4 The closed graph theorem 174
4.1 Grothendieck's theorem on closed subspaces of L^{p} 174
5 Besicovitch sets 176
6 Exercises 181
7 Problems 185
Chapter 5 Rudiments of Probability Theory 188
1 Bernoulli trials 189
1.1 Coin flips 189
1.2 The case N = ∞ 191
1.3 Behavior of SN as N → ∞, first results 194
1.4 Central limit theorem 195
1.5 Statement and proof of the theorem 197
1.6 Random series 199
1.7 Random Fourier series 202
1.8 Bernoulli trials 204
2 Sums of independent random variables 205
2.1 Law of large numbers and ergodic theorem 205
2.2 The role of martingales 208
2.3 The zeroone law 215
2.4 The central limit theorem 215
2.5 Random variables with values in Rd 220
2.6 Random walks 222
3 Exercises 227
4 Problems 235
Chapter 6 An Introduction to Brownian Motion 238
1 The Framework 239
2 Technical Preliminaries 241
3 Construction of Brownian motion 246
4 Some further properties of Brownian motion 251
5 Stopping times and the strong Markov property 253
5.1 Stopping times and the Blumenthal zeroone law 254
5.2 The strong Markov property 258
5.3 Other forms of the strong Markov Property 260
6 Solution of the Dirichlet problem 264
7 Exercises 268
8 Problems 273
Chapter 7 A Glimpse into Several Complex Variables 276
1 Elementary properties 276
2 Hartogs' phenomenon: an example 280
3 Hartogs' theorem: the inhomogeneous CauchyRiemann equations 283
4 A boundary version: the tangential CauchyRiemann equations 288
5 The Levi form 293
6 A maximum principle 296
7 Approximation and extension theorems 299
8 Appendix: The upper halfspace 307
8.1 Hardy space 308
8.2 Cauchy integral 311
8.3 Nonsolvability 313
9 Exercises 314
10 Problems 319
Chapter 8 Oscillatory Integrals in Fourier Analysis 321
1 An illustration 322
2 Oscillatory integrals 325
3 Fourier transform of surfacecarried measures 332
4 Return to the averaging operator 337
5 Restriction theorems 343
5.1 Radial functions 343
5.2 The problem 345
5.3 The theorem 345
6 Application to some dispersion equations 348
6.1 The Schrödinger equation 348
6.2 Another dispersion equation 352
6.3 The nonhomogeneous Schrödinger equation 355
6.4 A critical nonlinear dispersion equation 359
7 A look back at the Radon transform 363
7.1 A variant of the Radon transform 363
7.2 Rotational curvature 365
7.3 Oscillatory integrals 367
7.4 Dyadic decomposition 370
7.5 Almostorthogonal sums 373
7.6 Proof of Theorem 7.1 374
8 Counting lattice points 376
8.1 Averages of arithmetic functions 377
8.2 Poisson summation formula 379
8.3 Hyperbolic measure 384
8.4 Fourier transforms 389
8.5 A summation formula 392
9 Exercises 398
10 Problems 405
Notes and References 409
Bibliography 413
Symbol Glossary 417
Index 419