1 FOURIER SERIES ON THE CIRCLE
1.1 Motivation and Heuristics
1.1.1 Motivation from Physics
1.1.1.1 The Vibrating String
1.1.1.2 Heat Flow in Solids
1.1.2 Absolutely Convergent Trigonometric Series
1.1.3 *Examples of Factorial and Bessel Functions
1.1.4 Poisson Kernel Example
1.1.5 *Proof of Laplace's Method
1.1.6 *Nonabsolutely Convergent Trigonometric Series
1.2 Formulation of Fourier Series
1.2.1 Fourier Coefficients and Their Basic Properties
1.2.2 Fourier Series of Finite Measures
1.2.3 *Rates of Decay of Fourier Coefficients
1.2.3.1 Piecewise Smooth Functions
1.2.3.2 Fourier Characterization of Analytic Functions
1.2.4 Sine Integral
1.2.4.1 Other Proofs That Si(∞)=1
1.2.5 Pointwise Convergence Criteria
1.2.6 *Integration of Fourier Series
1.2.6.1 Convergence of Fourier Series of Measures
1.2.7 Riemann Localization Principle
1.2.8 Gibbs-Wilbraham Phenomenon
1.2.8.1 The General Case
1.3 Fourier Series in L2
1.3.1 Mean Square Approximation-Parseval's Theorem
1.3.2 *Application to the Isoperimetric Inequality
1.3.3 *Rates of Convergence in L2
1.3.3.1 Application to Absolutely-Convergent Fourier Series
1.4 Norm Convergence and Summability
1.4.1 Approximate Identifies
1.4.1.1 Almost-Everywhere Convergence of the Abel Means
1.4.2 Summability Matrices
1.4.3 Fejer Means of a Fourier Series
1.4.3.1 Wiener's Closure Theorem on the Circle
1.4.4 *Equidistribution Modulo One
1.4.5 *Hardy's Tauberian Theorem
1.5 Improved Trigonometric Approximation
1.5.1 Rates of Convergence in C(T)
1.5.2 Approximation with Fejer Means
1.5.3 *Jackson's Theorem
1.5.4 *Higher-Order Approximation
1.5.5 *Converse Theorems of Bemstein
1.6 Divergence of Fourier Series
1.6.1 The Example of du Bois-Reymond
1.6.2 Analysis via Lebesgue Constants
1.6.3 Divergence in the Space L1
1.7 *Appendix: Complements on Laplace's Method
1.7.0.1 First Variation on the Theme-Gaussian Approximation
1.7.0.2 Second Variation on the Theme-Improved Error Estimate
1.7.1 *Application to Bessel Functions
1.7.2 *The Local Limit Theorem of DeMoivre-Laplace
1.8 Appendix: Proof of the Uniform Boundedness Theorem
1.9 *Appendix: Higher-Order Bessel functions
1.10 Appendix: Cantor's Uniqueness Theorem
2 FOURIER TRANSFORMS ON THE LINE AND SPACE
2.1 Motivation and Heuristics
2.2 Basic Properties of the Fourier Transform
2.2.1 Riemann-Lebesgue Lemma
2.2.2 Approximate Identities and Gaussian Summability
2.2.2.1 Improved Approximate Identities for Pointwise Convergence
2.2.2.2 Application to the Fourier Transform
2.2.2.3 The n-Dimensional Poisson Kernel
2.2.3 Fourier Transforms of Tempered Distributions
2.2.4 *Characterization of the Gaussian Density
2.2.5 *Wiener's Density Theorem
2.3 Fourier Inversion in One Dimension
2.3.1 Dirichlet Kernel and Symmetric Partial Sums
2.3.2 Example of the Indicator Function
2.3.3 Gibbs-Wilbraham Phenomenon
2.3.4 Dini Convergence Theorem
2.3.4.1 Extension to Fourier's Single Integral
2.3.5 Smoothing Operations in R1-Averaging and Summability
2.3.6 Averaging and Weak Convergence
2.3.7 Cesbxo Summability
2.3.7.1 Approximation Properties of the Fejrr Kernel
2.3.8 Bernstein's Inequality
2.3.9 *One-Sided Fourier Integral Representation
2.3.9.1 Fourier Cosine Transform
2.3.9.2 Fourier Sine Transform
2.3.9.3 Generalized h-Transform
2.4 L2 Theory in Rn
2.4.1 Plancherel's Theorem
2.4.2 *Bernstein's Theorem for Fourier Transforms
2.4.3 The Uncertainty Principle
2.4.3.1 Uncertainty Principle on the Circle
2.4.4 Spectral Analysis of the Fourier Transform
