The Diestel-Leader graph DL(q, r) is the horocyclic product of the homogeneous trees with respective degrees q+1 and r+1. When q = r, it is the Cayley graph of the lamplighter group (wreath product) Zq≀Z with respect to a natural generating set. For the “Simple random walk” (SRW) operator on the latter group, Grigorchuk and Żuk and Dicks and Schick have determined the spectrum and the (on-diago...

The problem of finding the explicit Plancherel formulas for semisimple Lie groups has been solved completely in the case of complex semisimple Lie groups (see [3 (b)]). Moreover Harish-Chandra showed [3(f)] that the problem is solved also for a real semisimple Lie group having only one conjugate class of Cartan subgroups. In the case of real semisimple Lie groups with several conjugate classes ...

The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasi-regular representations of a semidirect product group G = A ⋊ H acting on L(A) requires the existence of a square-integrable function whose Plancherel transform satisfies Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable funct...

We discuss some existence conditions of the Shehu transform, we provide a Plancherel formula and also relate equicontinuity to exponential L2-equivanishing.

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let G be a time-frequency group. That is G = ⟨ Tk,Ml : k ∈ Z, l ∈ BZ ⟩ , where Tk, Ml are translations and modulations operators acting in L(R), and B is a non-singular matrix. We compute the Plancherel measure of the left regular representation of G which is...