Fourier series
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| Fourier series | |
|---|---|
| Name | Fourier series |
| Field | Mathematical analysis |
| Introduced | 19th century |
| Notable | Jean-Baptiste Joseph Fourier |
Fourier series are a way to represent a periodic function as an infinite sum of sines and cosines. They serve as a bridge between time-domain descriptions and frequency-domain analysis, underpinning methods used across science and engineering. Applications range from solving partial differential equations to signal processing and quantum mechanics.
Definition and Basic Concepts
A Fourier series expresses a periodic function f with period T as a sum of harmonically related trigonometric functions; the classical form uses coefficients multiplying cosines and sines or complex exponentials e^{inx}. Key definitions involve the fundamental frequency, harmonics, orthogonality relations of trigonometric functions, and inner products on function spaces such as L^2. Important figures and institutions associated with these foundational ideas include Jean-Baptiste Joseph Fourier, Joseph-Louis Lagrange, Pierre-Simon Laplace, Carl Friedrich Gauss, Augustin-Louis Cauchy, Niels Henrik Abel, and mathematical centers like the École Polytechnique, Collège de France, University of Göttingen, and École Normale Supérieure.
Convergence Theorems and Properties
Convergence results determine when the Fourier series of f converges to f or to an average of lateral limits. Classical theorems include Dirichlet's conditions, the Riemann–Lebesgue lemma, pointwise convergence under piecewise smoothness, uniform convergence for continuous and absolutely convergent series, and mean-square convergence governed by Parseval's identity. Deeper work involves theorems by mathematicians such as Dirichlet, Bernhard Riemann, Henri Lebesgue, Andrey Kolmogorov, Norbert Wiener, Stefan Banach, John von Neumann, and institutions like Collège de France and Institute for Advanced Study where harmonic analysis advanced. Counterexamples and pathological constructions come from contributors including Ulisse Dini, Georg Cantor, Paul du Bois-Reymond, Nikolai Luzin, and Salomon Bochner.
Fourier Coefficients and Computation
Fourier coefficients are computed via integrals of f against sine, cosine, or complex exponential bases over one period; formulas rely on orthogonality. Practical computation uses discrete approximations and the Fast Fourier Transform algorithm developed by Cooley–Tukey methods and championed in contexts involving James Cooley and John Tukey. Numerical implementations appear in software from organizations such as Bell Labs, Massachusetts Institute of Technology, IBM, Microsoft Research, and standards like those used by NASA. Analytical techniques exploit symmetries tied to contributions from Joseph Fourier, Pierre-Simon Laplace, Karl Weierstrass, Évariste Galois, and computational advances linked to Alan Turing and Claude Shannon.
Examples and Applications
Fourier series solve boundary-value problems for the heat equation and wave equation in domains studied by Jean-Baptiste Joseph Fourier, Lord Kelvin, George Gabriel Stokes, and Simeon Denis Poisson. Signal and image processing applications connect to work by Harry Nyquist, Claude Shannon, Dennis Gabor, Richard Hamming, and technology firms like Bell Labs and AT&T. In quantum mechanics, expansions relate to analyses by Erwin Schrödinger, Werner Heisenberg, Paul Dirac, and institutions such as Cavendish Laboratory and Institute for Advanced Study. Engineering uses include vibration analysis in projects involving Siemens, General Electric, Boeing, and Lockheed Martin. Applications in acoustics, optics, and communications trace to researchers like Lord Rayleigh, Maxwell, Heinrich Hertz, Guglielmo Marconi, and laboratories such as Philips Research Laboratories.
Extensions and Generalizations
Generalizations include Fourier transforms for nonperiodic functions, Fourier–Stieltjes series, Fourier integral representations, and expansions on manifolds and groups (harmonic analysis on Lie groups and compact groups). Wavelet transforms and time-frequency representations extend ideas developed in contexts involving Alfred Haar, Ingrid Daubechies, Yves Meyer, Stéphane Mallat, Jean Morlet, and institutions like Bell Labs. Noncommutative generalizations appear in work by Alain Connes and Israel Gelfand; distribution theory and Sobolev spaces bring in Laurent Schwartz and Sergei Sobolev. Connections to representation theory involve Emil Artin, Hermann Weyl, Harish-Chandra, and centers such as Institute for Advanced Study and Princeton University.
Historical Development and Notation
The historical arc began with Jean-Baptiste Joseph Fourier's study of heat propagation and his 1822 treatise, followed by rigorous analysis by Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann. Debates about convergence and function representation drew contributions from Joseph Liouville, Karl Weierstrass, Georg Cantor, Henri Lebesgue, and Paul du Bois-Reymond. Notation evolved from trigonometric sums to complex exponential bases influenced by Augustin-Louis Cauchy, Dirichlet, and later formalized in functional analysis by David Hilbert, Stefan Banach, and John von Neumann. Modern pedagogy and computational frameworks developed at institutions such as École Normale Supérieure, University of Cambridge, Harvard University, Massachusetts Institute of Technology, and Princeton University.