Fourier inversion theorem

Fourier inversion theorem
Name	Fourier inversion theorem
Field	Harmonic analysis
Introduced	19th century
Contributors	Jean-Baptiste Joseph Fourier, Joseph-Louis Lagrange, Adrien-Marie Legendre

Fourier inversion theorem The Fourier inversion theorem provides conditions under which a function can be recovered from its Fourier transform, establishing an equivalence between a function and its spectral representation. It lies at the heart of harmonic analysis, connects to the work of Jean-Baptiste Joseph Fourier, and underpins methods used by Joseph-Louis Lagrange, Carl Friedrich Gauss, and Bernhard Riemann in analysis and number theory. The theorem has deep ties to developments in École Polytechnique, Université de Paris, Princeton University, and modern institutions such as Institute for Advanced Study.

Statement

In one classical form, for a suitable function f on the real line, the Fourier transform F defined by an integral yields F(ξ) = ∫_{-∞}^{∞} f(x) e^{-2π i x ξ} dx, and the inversion formula reconstructs f almost everywhere by f(x) = ∫_{-∞}^{∞} F(ξ) e^{2π i x ξ} dξ. This formulation appears in treatments by Joseph Fourier and was made rigorous in the work of Bernhard Riemann, Henri Lebesgue, and Émile Borel. Equivalent statements are given for functions on Euclidean space, on the circle, and on locally compact abelian groups studied by Hermann Weyl and John von Neumann.

Conditions and Hypotheses

Different hypotheses allow different versions: for f in L^1(ℝ) with integrable transform, for f in L^2(ℝ) via the Plancherel theorem pioneered by Marcel Riesz and Frigyes Riesz, and for Schwartz functions considered by Laurent Schwartz. On the circle, conditions relate to Fourier series convergence as treated by Niels Henrik Abel and Sofia Kovalevskaya. For tempered distributions introduced by Laurent Schwartz the inversion holds in the sense of distributions, connecting to work of André Weil and Jean-Pierre Serre on automorphic forms. The Wiener algebra studied by Norbert Wiener gives another set of hypotheses ensuring uniform inversion. Historical refinements owe to studies by John Littlewood, G. H. Hardy, and Salomon Bochner.

Proofs

Proofs vary by setting: the L^1-L^∞ approach uses Fubini's theorem with ideas from Émile Borel and Henri Lebesgue; the L^2 proof uses the Plancherel identity and Hilbert space methods developed by David Hilbert and John von Neumann; Schwartz-space proofs employ mollifiers and approximation techniques related to S. R. Srinivasa Varadhan's ideas in analysis. Abstract harmonic analysis proofs for locally compact abelian groups invoke the Pontryagin duality theorem by Lev Pontryagin and expositions by Arne Beurling and Israel Gelfand. Kernel-based proofs use the Poisson summation formula linked to work by Siméon Denis Poisson and applications in number theory by G. H. Hardy and John Littlewood.

Applications

The inversion theorem underlies signal reconstruction in engineering contexts at institutions like Bell Labs and in technologies from AT&T to NASA. In quantum mechanics it supports the momentum-position duality central to Erwin Schrödinger and Werner Heisenberg formulations; in optics it appears in analyses by Lord Rayleigh and Augustin-Jean Fresnel. Number-theoretic applications connect to the explicit formula of Bernhard Riemann and spectral techniques used by Atle Selberg and Andrew Wiles. In partial differential equations, inversion techniques are used in the study of the heat equation by Joseph Fourier and in dispersive equations investigated by Terence Tao and Jean Bourgain. Computational methods such as the fast Fourier transform developed by James Cooley and John Tukey exploit inversion for efficient algorithms used by IBM and Intel.

Generalizations and Extensions

Generalizations extend to locally compact abelian groups via Pontryagin duality by Lev Pontryagin and to nonabelian harmonic analysis in works of Harish-Chandra and Israel Gelfand. The Paley–Wiener theorem attributed to Norbert Wiener and Raymond Paley characterizes transforms of compactly supported functions; the Titchmarsh convolution theorems by Edward Charles Titchmarsh refine support properties. Extensions to distributions and hyperfunctions follow from Laurent Schwartz and Mikio Sato. Multidimensional and manifold versions connect to analysis on Riemannian manifolds and representation theory of Lie groups as developed by Élie Cartan and Hermann Weyl.

Examples

- Gaussian: For f(x) = e^{-π x^2} the transform is itself, a fact used by Carl Friedrich Gauss and in heat-kernel studies by Joseph Fourier. - Rectangular pulse: The transform is a sinc function, classical in analyses by Oliver Heaviside and in communications work at Bell Labs. - Periodic functions: Fourier series inversion on the circle uses results by Jean-Baptiste Joseph Fourier and convergence studies by Niels Henrik Abel and Dirichlet. - Tempered distributions: The Dirac delta inversion is formalized in distribution theory by Laurent Schwartz and used in physics by Paul Dirac.

Category:Fourier analysis