Las leyes fundamentales de la estupidez humana

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. Book I Cover Half-Title Title Copyright Authors’ Dedications Foreword Preface to Book I Contents Chapter 1. The Genesis of Fourier Analysis 1.1 The vibrating string Simple harmonic motion Standing and traveling waves Harmonics and superposition of tones 1.1.1 Derivation of the wave equation 1.1.2 Solution to the wave equation Traveling waves Superposition of standing waves 1.1.3 Example: the plucked string 1.2 The heat equation 1.2.1 Derivation of the heat equation 1.2.2 Steady-state heat equation in the disc 1.3 Exercises 1.4 Problem Chapter 2. Basic Properties of Fourier Series 2.1 Examples and formulation of the problem Everywhere continuous functions Piecewise continuous functions Riemann integrable functions Functions on the circle 2.1.1 Main definitions and some examples 2.2 Uniqueness of Fourier series 2.3 Convolutions 2.4 Good kernels 2.5 Cesàro and Abel summability: applications to Fourier series 2.5.1 Cesàro means and summation 2.5.2 Fejér’s theorem 2.5.3 Abel means and summation 2.5.4 The Poisson kernel and Dirichlet’s problem in the unit disc 2.6 Exercises 2.7 Problems Chapter 3. Convergence of Fourier Series 3.1 Mean-square convergence of Fourier series 3.1.1 Vector spaces and inner products Preliminaries on vector spaces Two important examples 3.1.2 Proof of mean-square convergence 3.2 Return to pointwise convergence 3.2.1 A local result 3.2.2 A continuous function with diverging Fourier series 3.3 Exercises 3.4 Problems Chapter 4. Some Applications of Fourier Series 4.1 The isoperimetric inequality Curves, length and area Statement and proof of the isoperimetric inequality 4.2 Weyl’s equidistribution theorem The reals modulo the integers 4.3 A continuous but nowhere differentiable function 4.4 The heat equation on the circle 4.5 Exercises 4.6 Problems Chapter 5. The Fourier Transform on R 5.1 Elementary theory of the Fourier transform 5.1.1 Integration of functions on the real line 5.1.2 Definition of the Fourier transform 5.1.3 The Schwartz space 5.1.4 The Fourier transform on S The Gaussians as good kernels 5.1.5 The Fourier inversion 5.1.6 The Plancherel formula 5.1.7 Extension to functions of moderate decrease 5.1.8 The Weierstrass approximation theorem 5.2 Applications to some partial differential equations 5.2.1 The time-dependent heat equation on the real line 5.2.2 The steady-state heat equation in the upper half-plane 5.3 The Poisson summation formula 5.3.1 Theta and zeta functions 5.3.2 Heat kernels 5.3.3 Poisson kernels 5.4 The Heisenberg uncertainty principle 5.5 Exercises 5.6 Problems Chapter 6. The Fourier Transform on Rd 6.1 Preliminaries 6.1.1 Symmetries 6.1.2 Integration on Rd Polar coordinates 6.2 Elementary theory of the Fourier transform 6.3 The wave equation in Rd × R 6.3.1 Solution in terms of Fourier transforms 6.3.2 The wave equation in R3 × R Huygens principle 6.3.3 The wave equation in R2 × R: descent 6.4 Radial symmetry and Bessel functions 6.5 The Radon transform and some of its applications 6.5.1 The X-ray transform in R2 6.5.2 The Radon transform in R3 6.5.3 A note about plane waves 6.6 Exercises 6.7 Problems Chapter 7. Finite Fourier Analysis 7.1 Fourier analysis on Z(N) 7.1.1 The group Z(N) 7.1.2 Fourier inversion theorem and Plancherel identity on Z(N) 7.1.3 The fast Fourier transform 7.2 Fourier analysis on finite abelian groups 7.2.1 Abelian groups Examples of abelian groups The group Z∗(q) 7.2.2 Characters 7.2.3 The orthogonality relations 7.2.4 Characters as a total family 7.2.5 Fourier inversion and Plancherel formula 7.3 Exercises 7.4 Problems Chapter 8. Dirichlet’s Theorem 8.1 A little elementary number theory 8.1.1 The fundamental theorem of arithmetic 8.1.2 The infinitude of primes The zeta function and its Euler product 8.2 Dirichlet’s theorem 8.2.1 Fourier analysis, Dirichlet characters, and reduction of the 8.2.2 Dirichlet L-functions Historical digression 8.3 Proof of the theorem 8.3.1 Logarithms 8.3.2 L-functions 8.3.3 Non-vanishing of the L-function Case I: complex Dirichlet characters Case II: real Dirichlet characters 8.4 Exercises 8.5 Problems Appendix : Integration A.1 Definition of the Riemann integral A.1.1 Basic properties A.1.2 Sets of measure zero and discontinuities of integrable func- A.2 Multiple integrals A.2.1 The Riemann integral in Rd Definitions A.2.2 Repeated integrals A.2.3 The change of variables formula A.2.4 Spherical coordinates A.3 Improper integrals. Integration over Rd A.3.1 Integration of functions of moderate decrease A.3.2 Repeated integrals A.3.3 Spherical coordinates Notes and References Bibliography Symbol Glossary Index Book II Cover Half-Title Title Copyright Authors’ Dedications Foreword Contents Introduction Chapter 1. Preliminaries to Complex Analysis 1.1 Complex numbers and the complex plane 1.1.1 Basic properties 1.1.2 Convergence 1.1.3 Sets in the complex plane 1.2 Functions on the complex plane 1.2.1 Continuous functions 1.2.2 Holomorphic functions Complex-valued functions as mappings 1.2.3 Power series 1.3 Integration along curves 1.4 Exercises Chapter 2. Cauchy’s Theorem and Its Applications 2.1 Goursat’s theorem 2.2 Local existence of primitives and Cauchy’s theorem in a disc 2.3 Evaluation of some integrals 2.4 Cauchy’s integral formulas 2.5 Further applications 2.5.1 Morera’s theorem 2.5.2 Sequences of holomorphic functions 2.5.3 Holomorphic functions defined in terms of integrals 2.5.4 Schwarz reflection principle 2.5.5 Runge’s approximation theorem 2.6 Exercises 2.7 Problems Chapter 3. Meromorphic Functions and the Logarithm 3.1 Zeros and poles 3.2 The residue formula 3.2.1 Examples 3.3 Singularities and meromorphic functions The Riemann sphere 3.4 The argument principle and applications 3.5 Homotopies and simply connected domains 3.6 The complex logarithm 3.7 Fourier series and harmonic functions 3.8 Exercises 3.9 Problems Chapter 4. The Fourier Transform 4.1 The class F 4.2 Action of the Fourier transform on F 4.3 Paley-Wiener theorem 4.4 Exercises 4.5 Problems Chapter 5. Entire Functions 5.1 Jensen’s formula 5.2 Functions of finite order 5.3 Infinite products 5.3.1 Generalities 5.3.2 Example: the product formula for the sine function 5.4 Weierstrass infinite products 5.5 Hadamard’s factorization theorem Main lemmas Proof of Hadamard’s theorem 5.6 Exercises 5.7 Problems Chapter 6. The Gamma and Zeta Functions 6.1 The gamma function 6.1.1 Analytic continuation 6.1.2 Further properties of Γ 6.2 The zeta function 6.2.1 Functional equation and analytic continuation 6.3 Exercises 6.4 Problems Chapter 7. The Zeta Function and Prime Number Theorem 7.1 Zeros of the zeta function 7.1.1 Estimates for 1/ζ(s) 7.2 Reduction to the functions ψ and ψ1 7.2.1 Proof of the asymptotics for ψ1 Note on interchanging double sums 7.3 Exercises 7.4 Problems Chapter 8. Conformal Mappings 8.1 Conformal equivalence and examples 8.1.1 The disc and upper half-plane 8.1.2 Further examples 8.1.3 The Dirichlet problem in a strip Remarks about the Dirichlet problem 8.2 The Schwarz lemma; automorphisms of the disc and upper half-plane 8.2.1 Automorphisms of the disc 8.2.2 Automorphisms of the upper half-plane 8.3 The Riemann mapping theorem 8.3.1 Necessary conditions and statement of the theorem 8.3.2 Montel’s theorem 8.3.3 Proof of the Riemann mapping theorem 8.4 Conformal mappings onto polygons 8.4.1 Some examples 8.4.2 The Schwarz-Christoffel integral 8.4.3 Boundary behavior 8.4.4 The mapping formula 8.4.5 Return to elliptic integrals 8.5 Exercises 8.6 Problems Chapter 9. An Introduction to Elliptic Functions 9.1 Elliptic functions 9.1.1 Liouville’s theorems 9.1.2 The Weierstrass ℘ function An elliptic function of order two Properties of ℘ 9.2 The modular character of elliptic functions and Eisenstein series 9.2.1 Eisenstein series 9.2.2 Eisenstein series and divisor functions 9.3 Exercises 9.4 Problems Chapter 10. Applications of Theta Functions 10.1 Product formula for the Jacobi theta function 10.1.1 Further transformation laws 10.2 Generating functions 10.3 The theorems about sums of squares 10.3.1 The two-squares theorem 10.3.2 The four-squares theorem Statement of the theorem 10.4 Exercises 10.5 Problems Appendix A: Asymptotics A.1 Bessel functions A.2 Laplace’s method; Stirling’s formula A.3 The Airy function A.4 The partition function A.5 Problems Appendix B: Simple Connectivity and Jordan Curve Theorem B.1 Equivalent formulations of simple connectivity Winding numbers B.2 The Jordan curve theorem Proof of Theorem 2.1 Proof of Theorem 2.2 B.2.1 Proof of a general form of Cauchy’s theorem Notes and References Bibliography Symbol Glossary Index Book III Cover Half-Title Title Copyright Authors’ Dedications Foreword Contents Introduction 0.1 Fourier series: completion 0.2 Limits of continuous functions 0.3 Length of curves 0.4 Differentiation and integration 0.5 The problem of measure Chapter 1. Measure Theory 1.1 Preliminaries Open, closed, and compact sets Rectangles and cubes The Cantor set 1.2 The exterior measure Properties of the exterior measure 1.3 Measurable sets and the Lebesgue measure Invariance properties of Lebesgue measure Construction of a non-measurable set Axiom of choice 1.4 Measurable functions 1.4.1 Definition and basic properties 1.4.2 Approximation by simple functions or step functions 1.4.3 Littlewood’s three principles 1.5* The Brunn-Minkowski inequality 1.6 Exercises 1.7 Problems Chapter 2. Integration Theory 2.1 The Lebesgue integral: basic properties and convergence theorems Stage one: simple functions Stage two: bounded functions supported on a set of finite measure Return to Riemann integrable functions Stage three: non-negative functions Stage four: general case Complex-valued functions 2.2 The space L1 of integrable functions Invariance Properties Translations and continuity 2.3 Fubini’s theorem 2.3.1 Statement and proof of the theorem 2.3.2 Applications of Fubini’s theorem 2.4* A Fourier inversion formula 2.5 Exercises 2.6 Problems Chapter 3. Differentiation and Integration 3.1 Differentiation of the integral 3.1.1 The Hardy-Littlewood maximal function 3.1.2 The Lebesgue differentiation theorem 3.2 Good kernels and approximations to the identity 3.3 Differentiability of functions 3.3.1 Functions of bounded variation The Cantor-Lebesgue function 3.3.2 Absolutely continuous functions 3.3.3 Differentiability of jump functions 3.4 Rectifiable curves and the isoperimetric inequality 4.1* Minkowski content of a curve 4.2* Isoperimetric inequality 3.5 Exercises 3.6 Problems Chapter 4. Hilbert Spaces: An Introduction 4.1 The Hilbert space L2 4.2 Hilbert spaces 4.2.1 Orthogonality 4.2.2 Unitary mappings 4.2.3 Pre-Hilbert spaces 4.3 Fourier series and Fatou’s theorem 4.3.1 Fatou’s theorem 4.4 Closed subspaces and orthogonal projections 4.5 Linear transformations 4.5.1 Linear functionals and the Riesz representation theorem 4.5.2 Adjoints 4.5.3 Examples Infinite diagonal matrix Integral operators, and in particular, Hilbert-Schmidt operators 4.6 Compact operators 4.7 Exercises 4.8 Problems Chapter 5. Hilbert Spaces: Several Examples 5.1 The Fourier transform on L2 5.2 The Hardy space of the upper half-plane 5.3 Constant coefficient partial differential equations 5.3.1 Weak solutions 5.3.2 The main theorem and key estimate Proof of the main estimate 5.4* The Dirichlet principle 5.4.1 Harmonic functions The converse property 5.4.2 The boundary value problem and Dirichlet’s principle The two-dimensional theorem 5.5 Exercises 5.6 Problems Chapter 6. Abstract Measure and Integration Theory 6.1 Abstract measure spaces 6.1.1 Exterior measures and Carathéodory’s theorem 6.1.2 Metric exterior measures 6.1.3 The extension theorem 6.2 Integration on a measure space Measurable functions Definition and main properties of the integral The spaces L1(X, μ) and L2(X, μ) 6.3 Examples 6.3.1 Product measures and a general Fubini theorem 6.3.2 Integration formula for polar coordinates 6.3.3 Borel measures on R and the Lebesgue-Stieltjes integral 6.4 Absolute continuity of measures 6.4.1 Signed measures 6.4.2 Absolute continuity Mutually singular and absolutely continuous measures 6.5* Ergodic theorems 6.5.1 Mean ergodic theorem 6.5.2 Maximal ergodic theorem 6.5.3 Pointwise ergodic theorem 6.5.4 Ergodic measure-preserving transformations a) Rotations of the circle b) The doubling mapping 6.6* Appendix: the spectral theorem 6.6.1 Statement of the theorem 6.6.2 Positive operators 6.6.3 Proof of the theorem 6.6.4 Spectrum 6.7 Exercises 6.8 Problems Chapter 7. Hausdorff Measure and Fractals 7.1 Hausdorff measure 7.2 Hausdorff dimension 7.2.1 Examples The Cantor set Rectifiable curves The Sierpinski triangle The von Koch curve 7.2.2 Self-similarity 7.3 Space-filling curves 7.3.1 Quartic intervals and dyadic squares 7.3.2 Dyadic correspondence 7.3.3 Construction of the Peano mapping 7.4* Besicovitch sets and regularity 7.4.1 The Radon transform 7.4.2 Regularity of sets when d ≥ 3 7.4.3 Besicovitch sets have dimension 2 7.4.4 Construction of a Besicovitch set The proof that m(C + λC) = 0 for a.e. λ 7.5 Exercises 7.6 Problems Notes and References Bibliography Symbol Glossary Index Book IV Cover Half-Title Title Copyright Authors’ Dedications Foreword Contents Preface to Book IV Chapter 1. Lp Spaces and Banach Spaces 1.1 Lp spaces 1.1.1 The Hölder and Minkowski inequalities 1.1.2 Completeness of Lp 1.1.3 Further remarks 1.2 The case p = ∞ 1.3 Banach spaces 1.3.1 Examples 1.3.2 Linear functionals and the dual of a Banach space 1.4 The dual space of Lp when 1 ≤ p < ∞ 1.5 More about linear functionals 1.5.1 Separation of convex sets 1.5.2 The Hahn-Banach Theorem 1.5.3 Some consequences 1.5.4 The problem of measure 1.6 Complex Lp and Banach spaces 1.7 Appendix: The dual of C(X) 1.7.1 The case of positive linear functionals 1.7.2 The main result 1.7.3 An extension 1.8 Exercises 1.9 Problems Chapter 2. Lp Spaces in Harmonic Analysis 2.1 Early Motivations 2.2 The Riesz interpolation theorem 2.2.1 Some examples 2.3 The Lp theory of the Hilbert transform 2.3.1 The L2 formalism 2.3.2 The Lp theorem 2.3.3 Proof of Theorem 3.2 2.4 The maximal function and weak-type estimates 2.4.1 The Lp inequality Distribution function 2.5 The Hardy space H1 r 2.5.1 Atomic decomposition of H1 2.5.2 An alternative definition of H1 2.5.3 Application to the Hilbert transform 2.6 The space H1 r and maximal functions 2.6.1 The space BMO 2.7 Exercises 2.8 Problems Chapter 3. Distributions: Generalized Functions 3.1 Elementary properties 3.1.1 Definitions 3.1.2 Operations on distributions 3.1.3 Supports of distributions 3.1.4 Tempered distributions 3.1.5 Fourier transform 3.1.6 Distributions with point supports 3.2 Important examples of distributions 3.2.1 The Hilbert transform and pv( 1 3.2.2 Homogeneous distributions 3.2.3 Fundamental solutions 3.2.4 Fundamental solution to general partial differential equa- 3.2.5 Parametrices and regularity for elliptic equations 3.3 Calderón-Zygmund distributions and Lp estimates 3.3.1 Defining properties 3.3.2 The Lp theory 3.4 Exercises 3.5 Problems Chapter 4. Applications of the Baire Category Theorem 4.1 The Baire category theorem 4.1.1 Continuity of the limit of a sequence of continuous functions 4.1.2 Continuous functions that are nowhere differentiable Proof of property (ii) 4.2 The uniform boundedness principle 4.2.1 Divergence of Fourier series 4.3 The open mapping theorem 4.3.1 Decay of Fourier coefficients of L1-functions 4.4 The closed graph theorem 4.4.1 Grothendieck’s theorem on closed subspaces of Lp 4.5 Besicovitch sets 4.6 Exercises 4.7 Problems Chapter 5. Rudiments of Probability Theory 5.1 Bernoulli trials 5.1.1 Coin flips 5.1.2 The case N = ∞ 5.1.3 Behavior of SN as N → ∞, first results 5.1.4 Central limit theorem 5.1.5 Statement and proof of the theorem 5.1.6 Random series 5.1.7 Random Fourier series 5.1.8 Bernoulli trials 5.2 Sums of independent random variables 5.2.1 Law of large numbers and ergodic theorem 5.2.2 The role of martingales 5.2.3 The zero-one law 5.2.4 The central limit theorem 5.2.5 Random variables with values in Rd 5.2.6 Random walks 5.3 Exercises 5.4 Problems Chapter 6. An Introduction to Brownian Motion 6.1 The Framework 6.2 Technical Preliminaries 6.3 Construction of Brownian motion 6.4 Some further properties of Brownian motion 6.5 Stopping times and the strong Markov property 6.5.1 Stopping times and the Blumenthal zero-one law 6.5.2 The strong Markov property 6.5.3 Other forms of the strong Markov Property 6.6 Solution of the Dirichlet problem 6.7 Exercises 6.8 Problems Chapter 7. A Glimpse into Several Complex Variables 7.1 Elementary properties 7.2 Hartogs’ phenomenon: an example 7.3 Hartogs’ theorem: the inhomogeneous Cauchy-Riemann equations 7.4 A boundary version: the tangential Cauchy-Riemann equations 7.5 The Levi form 7.6 A maximum principle 7.7 Approximation and extension theorems 7.8 Appendix: The upper half-space 7.8.1 Hardy space 7.8.2 Cauchy integral 7.8.3 Non-solvability 7.9 Exercises 7.10 Problems Chapter 8. Oscillatory Integrals in Fourier Analysis 8.1 An illustration 8.2 Oscillatory integrals 8.3 Fourier transform of surface-carried measures 8.4 Return to the averaging operator 8.5 Restriction theorems 8.5.1 Radial functions 8.5.2 The problem 8.5.3 The theorem 8.6 Application to some dispersion equations 8.6.1 The Schrödinger equation 8.6.2 Another dispersion equation 8.6.3 The non-homogeneous Schrödinger equation 8.6.4 A critical non-linear dispersion equation 8.7 A look back at the Radon transform 8.7.1 A variant of the Radon transform 8.7.2 Rotational curvature 8.7.3 Oscillatory integrals 8.7.4 Dyadic decomposition 8.7.5 Almost-orthogonal sums 8.7.6 Proof of Theorem 7.1 8.8 Counting lattice points 8.8.1 Averages of arithmetic functions 8.8.2 Poisson summation formula 8.8.3 Hyperbolic measure 8.8.4 Fourier transforms 8.8.5 A summation formula 8.9 Exercises 8.10 Problems Notes and References Bibliography Symbol Glossary Index This First Volume, A Three-part Introduction To The Subject, Is Intended For Students With A Beginning Knowledge Of Mathematical Analysis Who Are Motivated To Discover The Ideas That Shape Fourier Analysis. It Begins With The Simple Conviction That Fourier Arrived At In The Early Nineteenth Century When Studying Problems In The Physical Sciences--that An Arbitrary Function Can Be Written As An Infinite Sum Of The Most Basic Trigonometric Functions. The Genesis Of Fourier Analysis -- The Vibrating String -- Derivation Of The Wave Equation -- Solution To The Wave Equation -- Example: The Plucked String -- The Heat Equation -- Derivation Of The Heat Equation -- Steady-state Heat Equation In The Disc -- Exercises -- Problem -- Basic Properties Of Fourier Series -- Examples And Formulation Of The Problem -- Main Definitions And Some Examples -- Uniqueness Of Fourier Series -- Convulusions -- Good Kernels -- Cesaro And Abel Summability: Applications To Fourier Series -- Cesaro Means And Summation -- Fejer's Theorem -- Abel Means And Summation -- The Poisson Kernel And Dirichlet's Problem In The Unit Disc -- Exercises -- Problems -- Convergence Of Fourier Series -- Mean-square Convergence Of Fourier Series -- Vector Spaces And Inner Products -- Proof Of Mean-square Convergence -- Return To Pointwise Convergence -- A Local Result -- A Continuous Function With Diverging Fourier Series -- Exercises -- Problems --^ Some Applications Of Fourier Series -- The Isoperimetric Inequality -- Weyl's Equidistribution Theorem -- A Continuous But Nowhere Differentiable Function -- The Heat Equation On The Circle -- Exercises -- Problems -- The Fourier Transform On R -- Elementary Theory Of The Fourier Transform -- Integration Of Functions On The Real Line -- Definition Of The Fourier Transform -- The Schwartz Space -- The Fourier Transform On S -- The Fourier Inversion -- The Plancherel Formula -- Extension To Functions Of Moderate Decrease -- The Weierstrass Approximation Theorem -- Applications To Some Partial Differential Equations -- The Time-dependent Heat Equation On The Real Line -- The Steady-state Heat Equation In The Upper Half-plane -- The Poisson Summation Formula -- Theta And Zeta Functions -- Heat Kernels -- Poisson Kernels -- The Heisenberg Uncertainty Principle -- Exercises -- Problems -- The Fourier Transform On Rd -- Preliminaries -- Symmetries -- Integration On Rd --^ Elementary Theory Of The Fourier Transform -- The Wave Equation In Rd X R -- Solution In Terms Of Fourier Transforms -- The Wave Equation In R3 X R -- The Wave Equation In Ir2 X R: Descent -- Radial Symmetry And Bessel Functions -- The Radon Transform And Some Of Its Applications -- The X-ray Transform In R2 -- The Radon Transform In R3 -- A Note About Plane Waves -- Exercises -- Problems -- Finite Fourier Analysis -- Fourier Analysis On Z(n) -- The Group Z(n) -- Fourier Inversion Theorem And Plancherel Identity On Z(n) -- The Fast Fourier Transform -- Fourier Analysis On Finite Abelian Groups -- Abelian Groups -- Characters -- The Orthogonality Relations -- Characters As A Total Family -- Fourier Inversion And Plancherel Formula -- Exercises -- Problems -- Dirichlet's Theorem -- A Little Elementary Number Theory -- The Fundamental Theorem Of Arithmetic -- The Infinitude Of Primes -- Dirichlet's Theorem -- Fourier Analysis, Dirichlet Characters, And Reduction Of The Theorem --^ Dirichlet L-functions -- Proof Of The Theorem -- Logarithms -- L-functions -- Non-vanishing Of The L-function -- Exercises -- Problems -- Appendix: Integration -- Definition Of The Riemann Integral -- Basic Properties -- Sets Of Measure Zero And Discontinuities Of Integrable Functions -- Multiple Integrals -- The Riemann Integral In Rd -- Repeated Integrals -- The Change Of Variables Formula -- Spherical Coordinates -- Improper Integrals. Integration Over Rd -- Integration Of Functions Of Moderate Decrease -- Repeated Integrals -- Spherical Coordinates -- Notes And References -- Bibliography -- Symbol Glossary. Elias M. Stein & Rami Shakarchi. Includes Bibliographical References (p. [301]-303) And Index. Real Analysis Is The Third Volume In The Princeton Lectures In Analysis, A Series Of Four Textbooks That Aim To Present, In An Integrated Manner, The Core Areas Of Analysis. Here The Focus Is On The Development Of Measure And Integration Theory, Differentiation And Integration, Hilbert Spaces, And Hausdorff Measure And Fractals. This Book Reflects The Objective Of The Series As A Whole: To Make Plain The Organic Unity That Exists Between The Various Parts Of The Subject, And To Illustrate The Wide Applicability Of Ideas Of Analysis To Other Fields Of Mathematics And Science. After Setting Forth The Basic Facts Of Measure Theory, Lebesgue Integration, And Differentiation On Euclidian Spaces, The Authors Move To The Elements Of Hilbert Space, Via The L2 Theory. They Next Present Basic Illustrations Of These Concepts From Fourier Analysis, Partial Differential Equations, And Complex Analysis. The Final Part Of The Book Introduces The Reader To The Fascinating Subject Of Fractional-dimensional Sets, Including Hausdorff Measure, Self-replicating Sets, Space-filling Curves, And Besicovitch Sets. Each Chapter Has A Series Of Exercises, From The Relatively-easy To The More-complex, That Are Tied Directly To The Text. A Substantial Number Of Hints Encourage The Reader To Take On Even The More-challenging Exercises. As With The Other Volumes In The Series, Real Analysis Is Accessible To Students Interested In Such Diverse Disciplines As Mathematics, Physics, Engineering, And Finance, At Both The Undergraduate And Graduate Levels. 1. Measure Theory -- 2. Integration Theory -- 3. Differentiation And Integration -- 4. Hilbert Spaces: An Introduction -- 5. Hilbert Spaces: Several Examples -- 6. Abstract Measure And Integration Theory -- 7. Hausdorff Measure And Fractals. Elias M. Stein & Rami Shakarchi. Includes Bibliographical References (p. 389-393) And Index. With this second volume we enter the intriguing world of complex analysis From the first theorems on the elegance and sweep of the results is evident The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex From there one proceeds to the main properties of holomorphic functions whose proofs are generally short and quite illuminating the Cauchy theorems residues analytic continuation the argument principle With this background the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics the Fourier transform treated by contour integration the zeta function and the prime number theorem and an introduction to elliptic functions culminating in their application to combinatorics and number theory Thoroughly developing a subject with many ramifications while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis Complex Analysis will be welcomed by students of mathematics physics engineering and other sciences The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them Numerous examples and applications throughout its four planned volumes of which Complex Analysis is the second highlight the far reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in depth considerations of complex analysis measure and integration theory and Hilbert spaces and finally further topics such as functional analysis distributions and elements of probability theory "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. A comprehensive and authoritative text that treats some of the main topics of modern analysis. A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables. Key results in each area discussed in relation to other areas of mathematics. Highlights the organic unity of large areas of analysis traditionally split into subfields. Interesting exercises and problems illustrate ideas. Clear proofs provided" -- "This book covers such topics as Lp spaces, distributions, Baire category, probability theory and Brownian motion, several complex variables and oscillatory integrals in Fourier analysis. The authors focus on key results in each area, highlighting their importance and the organic unity of the subject"-- "Real Analysis" is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, "Real Analysis" is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis:

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.

• A comprehensive and authoritative text that treats some of the main topics of modern analysis
• A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables
• Key results in each area discussed in relation to other areas of mathematics
• Highlights the organic unity of large areas of analysis traditionally split into subfields
• Interesting exercises and problems illustrate ideas
• Clear proofs provided

"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. A comprehensive and authoritative text that treats some of the main topics of modern analysis. A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables. Key results in each area discussed in relation to other areas of mathematics. Highlights the organic unity of large areas of analysis traditionally split into subfields. Interesting exercises and problems illustrate ideas. Clear proofs provided"-- Résumé de l'éditeur "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. A comprehensive and authoritative text that treats some of the main topics of modern analysis. A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables. Key results in each area discussed in relation to other areas of mathematics. Highlights the organic unity of large areas of analysis traditionally split into subfields. Interesting exercises and problems illustrate ideas. Clear proofs provided"-- Provided by publisher