Introduction to Harmonic Analysis (Student Mathematical Library)

✍ Scribed by Ricardo A. Saenz

• Publisher: American Mathematical Society
• Year: 1996
• Tongue: English
• Leaves: 296
• Category: Library

✦ Synopsis

This book gives a self-contained introduction to the modern ideas and problems of harmonic analysis. Intended for third- and fourth-year undergraduates, the book only requires basic knowledge of real analysis, and covers necessary background in measure theory, Lebesgue integration and approximation theorems. The book motivates the study of harmonic functions by describing the Dirichlet problem, and discussing examples such as solutions to the heat equation in equilibrium, the real and imaginary parts of holomorphic functions, and the minimizing functions of energy. It then leads students through an in-depth study of the boundary behavior of harmonic functions and finishes by developing the theory of harmonic functions defined on fractals domains. The book is designed as a textbook for an introductory course on classical harmonic analysis, or for a course on analysis on fractals. Each chapter contains exercises, and bibliographic and historical notes. The book can also be used as a supplemental text or for self-study.

✦ Table of Contents

Introduction to Harmonic Analysis
Half-title Page
Title Page
Copyright
Contents
IAS/Park City Mathematics Institute
Preface
Chapter 1. Motivation and preliminaries
1.1. The heat equation in equilibrium
1.2. Holomorphic functions
1.3. Know thy calculus
1.4. The Dirichlet principle
Exercises
Chapter 2. Basic properties
2.1. The mean value property
2.2. The maximum principle
2.3. Poisson kernel and Poisson integrals in the ball
2.4. Isolated singularities
Exercises
Notes
Chapter 3. Fourier series
3.1. Separation of variables
3.2. Fourier series
3.3. Abel means and Poisson integrals
3.4. Absolute convergence
3.5. Fejér’s theorem
3.6. Mean-square convergence
3.7. Convergence for continuous functions
Exercises
Notes
Chapter 4. Poisson kernel in the half-space
4.1. The Poisson kernel in the half-space
4.2. Poisson integrals in the half-space
4.3. Boundary limits
Exercises
Notes
Chapter 5. Measure theory in Euclidean space
5.1. The need for an integration theory
5.2. Outer measure in Euclidean space
5.3. Measurable sets and measure
5.4. Measurable functions
Exercises
Notes
Chapter 6. Lebesgue integral and Lebesgue spaces
6.1. Integration of measurable functions
6.2. Fubini’s theorem
6.3. The Lebesgue space 𝐿¹
6.4. The Lebesgue space 𝐿²
Exercises
Notes
Chapter 7. Maximal functions
7.1. Indefinite integrals and averages
7.2. The Hardy–Littlewood maximal function
7.3. The Lebesgue differentiation theorem
7.4. Boundary limits of harmonic functions
Exercises
Notes
Chapter 8. Fourier transform
8.1. Integrable functions
8.2. The Fourier inversion formula
8.3. Mean-square convergence
Exercises
Notes
Chapter 9. Hilbert transform
9.1. The conjugate function
9.2. Mean-square convergence
9.3. The Hilbert transform of integrable functions
9.4. Convergence in measure
Exercises
Notes
Chapter 10. Mathematics of fractals
10.1. Hausdorff dimension
10.2. Self-similar sets
Exercises
Notes
Chapter 11. The Laplacian on the Sierpiński gasket
11.1. Discrete energy on the interval
11.2. Harmonic structure on the Sierpiński gasket
11.3. The Laplacian on the Sierpiński gasket
Exercises
Notes
Chapter 12. Eigenfunctions of the Laplacian
12.1. Discrete eigenfunctions on the interval
12.2. Discrete eigenfunctions on the Sierpiński gasket
12.3. Dirichlet eigenfunctions
Exercises
Notes
Chapter 13. Harmonic functions on post-critically finite sets
13.1. Post-critically finite sets
13.2. Harmonic structures and discrete energy
13.3. Discrete Laplacians
13.4. The Laplacian on a PCF set
Exercises
Notes
Appendix A. Some results from real analysis
A.1. The real line
A.2. Topology
A.3. Riemann integration
A.4. The Euclidean space
A.5. Complete metric spaces
Acknowledgments
Bibliography
Index
Published Titles in this Subseries