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

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 250
Author(s):  
Frédéric Barbaresco ◽  
Jean-Pierre Gazeau
Keyword(s):  
Fourier Analysis ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Coherent States ◽  
Geometric Theory ◽  
Plancherel Formula ◽  
Group Representations ◽  
Modern Development ◽  
Xxist Century

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.

COHERENT STATES AND THEIR GENERALIZATIONS: A MATHEMATICAL OVERVIEW

1995 ◽  
Vol 07 (07) ◽  
pp. 1013-1104 ◽  
Author(s):  
S. TWAREQUE ALI ◽  
J.-P. ANTOINE ◽  
J.-P. GAZEAU ◽  
U.A. MUELLER
Keyword(s):  
Lie Groups ◽  
Quantum Measurement ◽  
Coherent States ◽  
Mathematical Structure ◽  
Standard Theory ◽  
Quantum Level ◽  
Semisimple Lie Groups ◽  
Group Representations ◽  
Group Context ◽  
Continuous Frames

We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

scholarly journals Plancherel Formula for Complex Semisimple Lie Groups

1951 ◽  
Vol 37 (12) ◽  
pp. 813-818 ◽  
Author(s):  
Harish-Chandra
Keyword(s):  
Lie Groups ◽  
Plancherel Formula ◽  
Semisimple Lie Groups ◽  
Complex Semisimple

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals Functional Equations and Fourier Analysis

2013 ◽  
Vol 56 (1) ◽  
pp. 218-224 ◽  
Author(s):  
Dilian Yang
Keyword(s):  
Representation Theory ◽  
Fourier Analysis ◽  
Harmonic Analysis ◽  
Functional Equations ◽  
Compact Groups ◽  
Wilson Equation ◽  
D’Alembert Equation

AbstractBy exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation-on compact groups.

Coherent States for Nilpotent Lie Groups

1986 ◽  
pp. 119-125
Author(s):  
Askold Perelomov
Keyword(s):  
Lie Groups ◽  
Coherent States ◽  
Nilpotent Lie Groups
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