Table of Contents

1. Linear Spaces 1.1 Functions 1.2 Vectors 1.3 Linear Spaces 1.4 Finite-dimensional Linear Spaces 1.5 Infinite-dimensional Linear Spaces 2. Orthogonal Functions 2.1 Inner Products 2.2 Orthogonal Functions and Vectors 2.3 Orthogonal Sequences 2.4 Differential Operators 2.5 Integral Operators 2.6 Convolution and the Dirichlet Kernel 3. Fourier Series 3.1 Motivation 3.2 Definitions 3.3 Examples of Trigonometric Series 3.4 Sine and Cosine Series 3.5 The Gibbs Phenomenon 3.6 Local Convergence of Fourier Series 3.7 Uniform Convergence 3.8 Convergence of Fourier Series 3.9 Divergent Series 3.10 Generalized Functions 3.11 Practical Remarks 4. Legendre Polynomials and Bessel Functions 4.1 Partial Differential Equations 4.2 The Intuitive Meaning of the Laplacian Operator 4.3 Legendre Polynomials 4.4 Laplace's Equation in Spherical Coordinates 4.5 Spherical Harmonics 4.6 Bessel Functions 5. Heat and Temperature 5.1 Theory of Heat Conduction 5.2 Temperature of Plates 5.3 Temperature of Solids 5.4 Harmonic Functions 5.5 Existence Theorems 5.6 Heat Flow 6. Waves and Vibrations, Harmonic Analysis 6.1 The Vibrating String 6.2 The One-dimensional Wave Equation 6.3 The Weighted String 6.4 String with Variable Tension and Density 6.5 Vibrating Membranes 6.6 Waves in Two and Three Dimensions 6.7 The Fourier Integral 6.8 Algebraic Concepts in Analysis Supplementary Exercises Appendix. Functions on Groups Answers and Notes; Index