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 instrument’s 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 Fourier’s general prescription.
Fourier’s claim was so remarkable and counterintuitive 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 farranging applicability of Fourier’s method, thereby
opening up vast new realms of mathematics, physics, engineering, and elsewhere. Indeed,
Fourier’s discovery easily ranks in the “top ten” mathematical advances of all time, a list
that would also include Newton’s invention of the calculus, and Gauss and Riemann’s
differential geometry that, 70 years later, became the foundation of Einstein’s 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 CD’s and DVD’s, 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
ε
–
δ
definition of limit and continuity, convergence properties in function space, the modern
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2012
Peter J. Olver
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Fourier Series, Jean Baptiste Joseph Fourier, Peter J. Olver

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