Advanced Real Analysis 2nd Edition by Anthony W Knapp ISBN 978-0817643829

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ISBN 13: 978-0817643829

Author: Anthony W Knapp

 In the years since publication of the first editions of Basic Real Analysis and Advanced Real Analysis, many readers have reacted to the books by sending comments, suggestions, and corrections. They appreciated the overall compre-hensive nature of the books, associating this feature with the large number of problems that develop so many sidelights and applications of the theory. 
Along with the general comments and specific suggestions were corrections, and there were enough corrections to Basic Real Analysis to warrant a second edition. A second edition of Advanced Real Analysis was then needed for compatibility. As long as this volume was being changed, it seemed appropriate to add the two important topics that are listed below. 
For the first editions, the author granted a publishing license to Birkhäuser Boston that was limited to print media, leaving the question of electronic publi-cation unresolved. A major change with the second editions is that the question of electronic publication has now been resolved, and for each book a PDF file, called the “digital second edition,” is being made freely available to everyone worldwide for personal use. These files may be downloaded from the author’s own Web page and from elsewhere. 
The two important additions to Advanced Real Analysis are as follows: 
Five sections on the Central Limit Theorem and an introduction to statistical inference. This material appears at the end of Chapter IX, “Foundations of Probability.” Four sections deal with the Central Limit Theorem itself. The fifth section shows how the Central Limit Theorem arises in applications to statistics as a limiting case of the distribution of W. S. Gosset, also known as Student’s distribution. Gosset’s distribution plays a fundamental role in statistical inference. 
A chapter on the theory of wavelets, with some commentary on applications. This material occupies Chapter X, “Introduction to Wavelets.” Wavelets form another theory, beyond the frequency analysis of Fourier series and the Fourier transform, for decomposing functions of one or more variables into component parts that bring out hidden behavior of the functions. The theory was introduced in the 1980s and 1990s to bring together disparate applications in signal processing and related fields, and it has now reached a sufficient state of maturity that all mathematicians might benefit from some familiarity with it. More information about the content of the chapter appears in the Guide to the Reader on pages xviii-xxii.

Table of contents: 

I. INTRODUCTION TO BOUNDARY-VALUE PROBLEMS

1. Partial Differential Operators

2. Separation of Variables

3. Sturm–Liouville Theory

4. Problems

II. COMPACT SELF-ADJOINT OPERATORS

1. Compact Operators

2. Spectral Theorem for Compact Self-Adjoint Operators

3. Hilbert–Schmidt Theorem

4. Unitary Operators

5. Classes of Compact Operators

6. Problems

III. TOPICS IN EUCLIDEAN FOURIER ANALYSIS

1. Tempered Distributions

2. Weak Derivatives and Sobolev Spaces

3. Harmonic Functions

4. Hp Theory

5. Calderon–Zygmund Theorem

6. Applications of the Calderon–Zygmund Theorem

7. Multiple Fourier Series

8. Application to Traces of Integral Operators

9. Problems

IV. TOPICS IN FUNCTIONAL ANALYSIS

1. Topological Vector Spaces

2. C∞(U), Distributions, and Support

3. Weak and Weak-Star Topologies, Alaoglu’s Theorem

4. Stone Representation Theorem

5. Linear Functionals and Convex Sets

6. Locally Convex Spaces

7. Topology on C∞com(U)

8. Krein–Milman Theorem

9. Fixed-Point Theorems

10. Gelfand Transform for Commutative C∗ Algebras

11. Spectral Theorem for Bounded Self-Adjoint Operators

12. Problems

V. DISTRIBUTIONS

1. Continuity on Spaces of Smooth Functions

2. Elementary Operations on Distributions

3. Convolution of Distributions

4. Role of Fourier Transform

5. Fundamental Solution of Laplacian

6. Problems

VI. COMPACT AND LOCALLY COMPACT GROUPS

1. Topological Groups

2. Existence and Uniqueness of Haar Measure

3. Modular Function

4. Invariant Measures on Quotient Spaces

5. Convolution and Lp Spaces

6. Representations of Compact Groups

7. Peter–Weyl Theorem

8. Fourier Analysis Using Compact Groups

9. Problems

VII. ASPECTS OF PARTIAL DIFFERENTIAL EQUATIONS

1. Introduction via Cauchy Data

2. Orientation

3. Local Solvability in the Constant-Coefficient Case

4. Maximum Principle in the Elliptic Second-Order Case

5. Parametrices for Elliptic Equations with Constant Coefficients

6. Method of Pseudodifferential Operators

7. Problems

VIII. ANALYSIS ON MANIFOLDS

1. Differential Calculus on Smooth Manifolds

2. Vector Fields and Integral Curves

3. Identification Spaces

4. Vector Bundles

5. Distributions and Differential Operators on Manifolds

6. More about Euclidean Pseudodifferential Operators

7. Pseudodifferential Operators on Manifolds

8. Further Developments

9. Problems

IX. FOUNDATIONS OF PROBABILITY

1. Measure-Theoretic Foundations

2. Independent Random Variables

3. Kolmogorov Extension Theorem

4. Strong Law of Large Numbers

5. Convergence in Distribution

6. Portmanteau Lemma

7. Characteristic Functions

8. Levy Continuity Theorem

9. Central Limit Theorem

10. Statistical Inference and Gosset’s t Distribution

11. Problems

X. INTRODUCTION TO WAVELETS

1. Introduction

2. Haar Wavelet

3. Multiresolution Analysis

4. Shannon Wavelet

5. Construction of a Wavelet from a Scaling Function

6. Meyer Wavelets

7. Splines

8. Battle–Lemarie Wavelets

9. Daubechies Wavelets

10. Smoothness Questions

11. A Quick Introduction to Applications

12. Problems

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