CONTENT S
1 FOURIER SERIES ON THE CIRCLE 1
1.1 Motivation and Heuristics 1
1.1.1 Motivation from Physics 1
1.1.1.1 The Vibrating String 1
1.1.1.2 Heat Flow in Solids 2
1.1.2 Absolutely Convergent Trigonometric Series 3
1.1.3 * Examples of Factorial and Bessel Functions 6
1.1.4 Poisson Kernel Example 7
1.1.5 *Proof of Laplace's Method 9
1.1.6 *Nonabsolutely Convergent Trigonometric Series 11
1.2 Formulation of Fourier Series 13
1.2.1 Fourier Coefficients and Their Basic Properties 13
1.2.2 Fourier Series of Finite Measures 19
1.2.3 *Rates of Decay of Fourier Coefficients 20
1.2.3.1 Piecewise Smooth Functions 21
1.2.3.2 Fourier Characterization of Analytic Functions 22
1.2.4 Sine Integral 24
1.2.4.1 Other Proofs That Si(oo) = 1 24
1.2.5 Pointwise Convergence Criteria 25
1.2.6 ^Integration of Fourier Series 29
1.2.6.1 Convergence of Fourier Series of Measures 30
1.2.7 Riemann Localization Principle 31
1.2.8 Gibbs-Wilbraham Phenomenon 31
1.2.8.1 The General Case 34
1.3 Fourier Series in L2 35
1.3.1 Mean Square Approximation—Parseval's Theorem 35
1.3.2 * Application to the Isoperimetric Inequality 38
ix
