Introduction to Harmonic Analysis

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