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

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