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Displaying similar documents to “Spherical functions and spectral synthesis”

An application of shift operators to ordered symmetric spaces

Nils Byrial Andersen, Jérémie M. Unterberger (2002)

Annales de l’institut Fourier

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We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ℳ m , n of Cayley type, where m ∈ ℕ denotes the multiplicity of the short roots and n ∈ ℕ the rank of the symmetric space. For m even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ℳ m , n by a reduction to the rank 1 case. Finally we generalize our notions and results to B C n type root systems.

Spherical functions on ordered symmetric spaces

Jacques Faraut, Joachim Hilgert, Gestur Ólafsson (1994)

Annales de l'institut Fourier

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We define on an ordered semi simple symmetric space ℳ = G / H a family of spherical functions by an integral formula similar to the Harish-Chandra integral formula for spherical functions on a Riemannian symmetric space of non compact type. Associated with these spherical functions we define a spherical Laplace transform. This transform carries the composition product of invariant causal kernels onto the ordinary product. We invert this transform when G is a complex group, H a real form of G ,...

Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform

Detlev Poguntke (2010)

Colloquium Mathematicae

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For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple...

A Paley-Wiener theorem on NA harmonic spaces

Francesca Astengo, Bianca di Blasio (1999)

Colloquium Mathematicae

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Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.