To the Student

This book is about one big idea: You can synthesize a variety of complicated functions from pure sinusoids in much the same way that you produce a major chord by striking nearby C, E, G keys on a piano. A geometric version of this idea forms the basis for the ancient Hipparchus-Ptolemy model of planetary motion (Almagest, 2nd century see Fig. 1.2). It was Joseph Fourier (Analytical Theory of Heat, 1815), however, who developed modern methods for using trigonometric series and integrals as he studied the flow of heat in solids. Today, Fourier analysis is a highly evolved branch of mathematics with an incomparable range of applications and with an impact that is second to none (see Appendix 1). If you are a student in one of the mathematical, physical, or engineering sciences, you will almost certainly find it necessary to learn the elements of this subject. My goal in writing this book is to help you acquire a working knowledge of Fourier analysis early in your career.

If you have mastered the usual core courses in calculus and linear algebra, you have the maturity to follow the presentation without undue difficulty. A few of the proofs and more theoretical exercises require concepts (uniform continuity, uniform convergence, …) from an analysis or advanced calculus course. You may choose to skip over the difficult steps in such arguments and simply accept the stated results. The text has been designed so that you can do this without severely impacting your ability to learn the important ideas in the subsequent chapters.