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Unformatted text preview: Chapter 3 Fourier Series Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. Through his deep analytical investigations into the partial differential equations modeling 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. For example, consider the sound produced by a musical instrument, e.g., piano, violin, trumpet, or drum. Decomposing the signal into its trigonometric constituents reveals the fundamental frequencies (tones, overtones, etc.) that combine to produce the instruments distinctive timbre. This 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 Fouriers general prescription. Fouriers claim was so remarkable and counter-intuitive that most of the leading math- ematicians of the time did not believe him. Nevertheless, it was not long before scientists came to appreciate the power and far-ranging applicability of Fouriers method, thereby opening up vast new realms of mathematics, physics, engineering, and elsewhere. Indeed, Fouriers discovery easily ranks in the top ten mathematical advances of all time, a list that would also include Newtons invention of the calculus, and Gauss and Riemanns differential geometry that, 70 years later, became the foundation of Einsteins general rel- ativity. 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 engineer- ing are almost too numerous to catalogue: typing the word Fourier in the subject index of a modern science library will dramatically demonstrate just how ubiquitous these meth- ods 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 CDs and DVDs, movies, computer graphics, image processing, and fingerprint analysis and storage, are, in one way or another, founded upon the many ramifications of Fourier theory. 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 weapons in your mathematical arsenal. Mastery of the subject is unavoidable. Furthermore, a remarkably large fraction of modern mathematics rests on subsequent attempts to place Fourier series on a firm mathematical foundation. Thus, all of mod- ern analysis most basic analytical tools, including the definition of a function, the definition of limit and continuity, convergence properties in function space, the modern 1/19/12 52 c circlecopyrt 2012 Peter J. Olver theory of integration and measure, generalized functions such as the delta function, and...
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