Barbaresco Frédéric, Gazeau Jean-Pierre
Key Technology Domain PCC (Processing, Control & Cognition) Representative, Thales Land & Air Systems, Voie Pierre-Gilles de Gennes, F91470 Limours, France.
APC (UMR 7164), Department of Physics, Université Paris-Diderot, F75205 Paris, France.
Entropy (Basel). 2019 Mar 6;21(3):250. doi: 10.3390/e21030250.
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.
为庆祝1768年出生于法国欧塞尔的约瑟夫·傅里叶诞辰250周年，本MDPI特刊将探讨与傅里叶分析和傅里叶热方程相关的现代主题。傅里叶分析以约瑟夫·傅里叶命名，涉及经典的交换调和分析。20世纪傅里叶分析的现代发展探索了傅里叶公式和傅里叶 - 普兰谢尔公式在非交换调和分析中的推广，应用于局部紧非阿贝尔群。与此同时，相干态和小波理论已在李群上得到推广（通过将相干态与在齐性空间上平方可积的群表示相关联）。约瑟夫·傅里叶的名字也与热的数学研究密不可分。热方程的现代研究探索了黎曼流形、次黎曼流形和李群上经典扩散方程的几何扩展。对于一般体积形式（不一定与黎曼体积形式一致）的热方程在次黎曼几何中很有用，因为在某些情况下才存在规范体积。通过应用为统计力学扩展的几何力学工具，例如李群热力学，一种新的热几何理论正在兴起。
