On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

Micheli, Enrico De

doi:10.3390/math8020287

Cited by 2 publications

(2 citation statements)

References 20 publications

(45 reference statements)

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“…Recently, Enrico De Micheli [ 69 ] has introduced a Laplace-type transform (the so-called Spherical Laplace Transform) with a connection to the Non-Euclidean Fourier Transform in the sense of Helgason, and the principal series of the unitary representation of SU(1,1) .…”

Section: Covariant Gibbs Density By Souriau Thermodynamics For Poimentioning

confidence: 99%

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

Barbaresco¹

2020

Entropy

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In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

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Section: Covariant Gibbs Density By Souriau Thermodynamics For Poimentioning

confidence: 99%

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

Barbaresco¹

2020

Entropy

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“…=0 f e −i t (see Formulae (A3), (A4) and (A19)). Then, it is quite natural to introduce the complex plane C of the variable τ = t + iw, (t, w ∈ R), and consider in this plane the following domains (see also [12]). For ξ 0 0 we define: I 1a) and 1b).…”

Section: Holomorphic Extension Associated With the Trigonometrical Seriesmentioning

confidence: 99%

Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem

Micheli¹

2021

Symmetry

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In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves fℓ of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves fℓ satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function f˜(λ)∈C, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function f(θ)∈C (θ being the complexified scattering angle), which is analytic in a strip contained in the θ-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex cosθ-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the λ- and θ-planes. The function f(θ) is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density σ(μ) associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given.

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On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

Cited by 2 publications

References 20 publications

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem

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