Chapter 12 Fourier Series

Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his investigations into the partial differential equations modeling vibration and heat propagation in bodies, Fourier was led to claim that “every” function could be represented by an infinite series of elementary trigonometric functions — sines and cosines. As an example, consider the sound produced by a musical instrument, e.g., piano, violin, trumpet, oboe, or drum. Decomposing the signal into its trigonometric constituents reveals the fundamental frequencies (tones, overtones, etc.) that are combined to produce its distinctive timbre. The Fourier decomposition lies at the heart of modern electronic music; a synthesizer combines pure sine and cosine tones to reproduce the diverse sounds of instruments, both natural and artificial, according to Fourier’s general prescription. Fourier’s claim was so remarkable and unexpected that most of the leading mathematicians of the time did not believe him. Nevertheless, it was not long before scientists came to appreciate the power and far-ranging applicability of Fourier’s method, thereby opening up vast new realms of physics, engineering, and elsewhere, to mathematical analysis. Indeed, Fourier’s discovery easily ranks in the “top ten” mathematical advances of all time, a list that would include Newton’s invention of the calculus, and Gauss and Riemann’s establishment of differential geometry that, 70 years later, became the foundation of Einstein’s general relativity. Fourier analysis is an essential component of much of modern applied (and pure) mathematics. It forms an exceptionally powerful analytical tool for solving a broad range of partial differential equations. Applications in pure mathematics, physics and engineering are almost too numerous to catalogue — typing in “Fourier” in the subject index of a modern science library will dramatically demonstrate just how ubiquitous these methods are. Fourier analysis lies at the heart of signal processing, including audio, speech, images, videos, seismic data, radio transmissions, and so on. Many modern technological advances, including television, music CD’s and DVD’s, video movies, computer graphics, image processing, and fingerprint analysis and storage, are, in one way or another, founded upon the many ramifications of Fourier’s discovery. In your career as a mathematician, scientist or engineer, you will find that Fourier theory, like calculus and linear algebra, is one of the most basic and essential tools in your mathematical arsenal. Mastery of the subject is unavoidable. Furthermore, a remarkably large fraction of modern pure mathematics is the result of subsequent attempts to place Fourier series on a firm mathematical foundation. Thus, all of the student’s “favorite” analytical tools, including the definition of a function, the ε–δ definition of limit and continuity, convergence properties in function space, including uni-