scholarly journals Direct integral theory for weights, and the Plancherel formula

1974 ◽  
Vol 80 (3) ◽  
pp. 456-462 ◽  
Author(s):  
Colin E. Sutherland
Keyword(s):  
Integral Theory ◽  
Plancherel Formula ◽  
Direct Integral

scholarly journals The Plancherel formula for the horocycle space and generalizations

1996 ◽  
Vol 61 (1) ◽  
pp. 42-56 ◽  
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Harmonic Analysis ◽  
Regular Representation ◽  
Plancherel Formula ◽  
Intertwining Operators ◽  
Direct Integral ◽  
Regular Representations ◽  
Spectral Distributions ◽  
Theoretic Technique ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractThe Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

scholarly journals The Plancherel formula for the horocycle spaces and generalizations, II

1998 ◽  
Vol 65 (2) ◽  
pp. 194-211
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Homogeneous Spaces ◽  
Regular Representation ◽  
Plancherel Formula ◽  
New Paradigm ◽  
Direct Integral ◽  
Stability Group ◽  
Nuclear Operators ◽  
Infinite Multiplicity ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractThe Plancherel formula for various semisimple homogeneous spaces with non-reductive stability group is derived within the framework of the Bonnet Plancherel formula for the direct integral decomposition of a quasi-regular representation. These formulas represent a continuation of the author's program to establish a new paradigm for concrete Plancherel analysis on homogeneous spaces wherein the distinction between finite and infinite multiplicity is de-emphasized. One interesting feature of the paper is the computation of the Bonnet nuclear operators corresponding to certain exponential representations (roughly those induced from infinite-dimensional representations of a subgroup). Another feature is a natural realization of the direct integral decomposition over a canonical set of concrete irreducible representations, rather than over the unitary dual.

scholarly journals On the structure of polynomially normal operators

1984 ◽  
Vol 30 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Compact Operator ◽  
Normal Operator ◽  
Integral Theory ◽  
Direct Integral ◽  
Direct Sum ◽  
Essentially Normal ◽  
Normal Operators

We present some results concerning the structure of polynomially normal operators. It is shown, among other things, that if Tn is normal for some n > 1, then T is quasi–similar to a direct sum of a normal operator and a compact operator and if p(T) is normal with T essentially normal, then T can be written as the sum of a normal operator and a compact operator. Utilizing the direct integral theory of operators we finally show that if p(T) is normal and T*T commutes with T + T*, then T must be normal.

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.

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