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Entropy 2019, 21(3), 250; https://doi.org/10.3390/e21030250

Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

1
Key Technology Domain PCC (Processing, Control & Cognition) Representative, Thales Land & Air Systems, Voie Pierre-Gilles de Gennes, F91470 Limours, France
2
APC (UMR 7164), Department of Physics, Université Paris-Diderot, F75205 Paris, France
*
Author to whom correspondence should be addressed.
Received: 19 February 2019 / Revised: 27 February 2019 / Accepted: 27 February 2019 / Published: 6 March 2019
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Abstract

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics. View Full-Text
Keywords: harmonic analysis on abstract space; heat equation on manifolds and Lie Groups harmonic analysis on abstract space; heat equation on manifolds and Lie Groups
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Barbaresco, F.; Gazeau, J.-P. Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century. Entropy 2019, 21, 250.

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