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A Bochner type theorem for inductive limits of Gelfand pairs

Marouane Rabaoui (2008)

Annales de l’institut Fourier

In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs ( G , K ) defined as inductive limits of increasing sequences of Gelfand pairs ( G ( n ) , K ( n ) ) n 1 . By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element ϕ of the set 𝒫 ( G ) of K -biinvariant continuous functions of positive type on G .

A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree

Michael G. Cowling, Stefano Meda, Alberto G. Setti (2010)

Colloquium Mathematicae

We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.

A weighted Plancherel formula II. The case of the ball

Genkai Zhang (1992)

Studia Mathematica

The group SU(1,d) acts naturally on the Hilbert space L ² ( B d μ α ) ( α > - 1 ) , where B is the unit ball of d and d μ α the weighted measure ( 1 - | z | ² ) α d m ( z ) . It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

An analogue of Gutzmer's formula for Hermite expansions

S. Thangavelu (2008)

Studia Mathematica

We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of L²(ℝⁿ) under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.

Asymptotic spherical analysis on the Heisenberg group

Jacques Faraut (2010)

Colloquium Mathematicae

The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair. The main results...

Distances hilbertiennes invariantes sur un espace homogène

Jacques Faraut, Khelifa Harzallah (1974)

Annales de l'institut Fourier

Nous déterminons pour certains espaces homogènes X = G / K les distances invariantes qui proviennent d’un plongement de X dans un espace de Hilbert. Le carré d’une telle distance est un noyau de type négatif invariant dont nous donnons une représentation, c’est la formule de Lévy-Kinchine. Nous en déduisons que si G possède la propriété (T) de Kajdan une telle distance est toujours bornée.

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