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

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