Volume 6, Issue 2 - Special Issue: Geometric and harmonic analysis on homogeneous spaces and applications: Part I. Proceedings of the Third Tunisian-Japanese Conference, Hammamet, Tunisia, December 2013

Let ℍ 2 n +1 r be the reduced Heisenberg Lie group, G = ℍ 2 n +1 r × ℍ 2 n +1 r be the (4 n + 2)-dimensional Lie group and Δ G = {( x , x ) ∈ G : x ∈ ℍ 2 n +1 r } be the diagonal subgroup of G . Given any discontinuous subgroup Γ ⊂ G for G /Δ G , we provide a layering of the parameter space ℛ(Γ, G , Δ G ), which is shown to be endowed with a smooth manifold structure, we also show that the stability property holds. On the other hand, a local (and hence a global) rigidity theorem is obtained. That is, the parameter space ℛ(Γ, G , Δ G ) admits a rigid point if and only if Γ is finite and this is also equivalent to the fact that the deformation space is Hausdorff.

We complete here our recent work on explicit approximations of the Prolate Spheroidal Wave Functions (PSWFs) on the interval [-1,+1] and their associated spectra by pushing forward the methods in view of new results. We give in particular approximations of the ratio between large and small oscillations of PSWFs as well of the transition bandwidth for which the PSWF stops to take its largest values at the boundary of the interval. The aim of this work is two folds. On one hand, we prove a bunch of properties of the PSWFs and on the other hand, we illustrate the theoretical results by some numerical experiments.

Let G be a locally compact group. We denote by 𝒮𝒰ℬ(ℝ × G ) the hyperspace of closed subgroups of ℝ × G endowed with the Chabauty topology. In this paper, some general results about the connectedness of the hyperspace 𝒮𝒰ℬ(ℝ × G ) are established. The main result proved here is that if G is a prosolvable group, then 𝒮𝒰ℬ(ℝ × G ) is connected. We also prove a partial converse to the result just mentioned.

Let G be an exponential solvable Lie group with Lie algebra 𝔤 defined by a semi-direct product of ℝ and ℝ n . We study a holomorphically induced representation ρ of G from a real linear form f of 𝔤 and a 1-dimensional complex subalgebra 𝔥 of 𝔤 ℂ such that the space 𝔥 + 𝔥̅ generates 𝔤 ℂ . Under some assumptions of structure of the Lie algebra, we obtain a sufficient condition for non-triviality of ρ and decompose ρ into a multiplicity-free direct integral of irreducible representations.

We consider the transmutation operators V k and t V k associated with the Cherednik operators attached to the root system A 2 , called also in [Adv. Pure Appl. Math. 1 (2010), 293–323; Adv. Pure Appl. Math. 2 (2011), 223–246] the trigonometric Dunkl intertwining operator, and its dual. In this paper, we prove that the operators V k and t V k are positivity preserving and allows positive integral representations. In particular, we deduce that the Opdam–Cherednik kernel is positive definite.