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

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.

Applications of sinusoidal phase modulation in temporal optics to highlight some properties of the Fourier transform

“…Finally, these lab sessions also enabled us to discuss academic research currently under investigation by different groups. This shows that the analysis proposed by Joseph Fourier [55,56] (of whom 2018 marks the 250th anniversary of his birth in our region) remains of the most crucial importance in modern applications such as high-speed transmissions.…”

TERMOFLUXíMETRO: SISTEMA DE MONITORAMENTO ONLINE REFRATáRIO PARA CANAL DE CORRIDA NA CSN

“…O método desenvolvido compreende o cálculo do fluxo de calor e devido a isso a tecnologia é chamada de termofluxímetro. O fluxo de calor representa a taxa de energia transferida por unidade de tempo e é definido pela Lei de Fourier [6] apresentada na equação abaixo.…”