前辅文
0 Overview of Measure Theory and Functional Analysis
0.1 Pre-Basics
0.2 A Whirlwind Review of Measure Theory
0.3 The Elements of Banach Space Theory
0.4 Hilbert Space
0.5 Two Fundamental Principles of Functional Analysis
1 Fourier Series Basics
1.0 The Pre-History of Fourier Analysis
1.1 The Rudiments of Fourier Series
1.2 Summability of Fourier Series
1.3 A Quick Introduction to Summability Methods
1.4 Key Properties of Summability Kernels
1.5 Pointwise Convergence for Fourier Series
1.6 Norm Convergence of Partial Sums and the Hilbert Transform
2 The Fourier Transform
2.1 Basic Properties of the Fourier Transform
2.2 Invariance and Symmetry Properties of the Fourier Transform
2.3 Convolution and Fourier Inversion
2.4 The Uncertainty Principle
3 Multiple Fourier Series
3.1 Various Methods of Partial Summation
3.2 Examples of Different Types of Summation
3.3 Fourier Multipliers and the Summation of Series
3.4 Applications of the Fourier Multiplier Theorems to Summation of Multiple Trigonometric Series
3.5 The Multiplier Problem for the Ball
4 Spherical Harmonics
4.1 A New Look at Fourier Analysis in the Plane
4.2 Further Results on Spherical Harmonics
5 Fractional Integrals, Singular Integrals, and Hardy Spaces
5.1 Fractional Integrals and Other Elementary Operators
5.2 Prolegomena to Singular Integral Theory
5.3 An Aside on Integral Operators
5.4 A Look at Hardy Spaces in the Complex Plane
5.5 The Real-Variable Theory of Hardy Spaces
5.6 The Maximal-Function Characterization of Hardy Spaces
5.7 The Atomic Theory of Hardy Spaces
5.8 Ode to BMO
6 Modern Theories of Integral Operators
6.1 Spaces of Homogeneous Type
6.2 Integral Operators on a Space of Homogeneous Type
6.3 A New Look at Hardy Spaces
6.4 The T .1/ Theorem
7 Wavelets
7.1 Localization in the Time and Space Variables
7.2 Building a Custom Fourier Analysis
7.3 The Haar Basis
7.4 Some Illustrative Examples
7.5 Construction of a Wavelet Basis
8 A Retrospective
8.1 Fourier Analysis: An Historical Overview
Appendices and Ancillary Material
Appendix I, The Existence of Testing Functions and Their Density in L p
Appendix II, Schwartz Functions and the Fourier Transform
Appendix III, The Interpolation Theorems of Marcinkiewicz and Riesz-Thorin
Appendix IV, Hausdorff Measure and Surface Measure
Appendix V, Green’s Theorem
Appendix VI, The Banach-Alaoglu Theorem
Appendix VII, Expressing an Integral in Terms of the Distribution Function
Appendix VIII, The Stone-Weierstrass Theorem
Appendix IX, Landau’s O and o Notation
Table of Notation
Bibliography
Index
