Fourier Series and Orthogonal Functions: (Dover Books on Mathema 1.4tics)

This incisive text deftly combines both theory and practical example to introduce and explore Fourier series and orthogonal functions and applications of the Fourier method to the solution of boundary-value problems. Directed to advanced undergraduate and graduate students in mathematics as well as in physics and engineering, the book requires no prior knowledge of partial differential equations or advanced vector analysis. Students familiar with partial derivatives, multiple integrals, vectors, and elementary differential equations will find the text both accessible and challenging. The first three chapters of the book address linear spaces, orthogonal functions, and the Fourier series. Chapter 4 introduces Legendre polynomials and Bessel functions, and Chapter 5 takes up heat and temperature. The concluding Chapter 6 explores waves and vibrations and harmonic analysis. Several topics not usually found in undergraduate texts are included, among them summability theory, generalized functions, and spherical harmonics. Throughout the text are 570 exercises devised to encourage students to review what has been read and to apply the theory to specific problems. Those preparing for further study in functional analysis, abstract harmonic analysis, and quantum mechanics will find this book especially valuable for the rigorous preparation it provides. Professional engineers, physicists, and mathematicians seeking to extend their mathematical horizons will find it an invaluable reference as well.

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