We construct the following: a perfect non Dirichlet set every proper closed subset of which is Kronecker, A weak Kronecker set which is not an $R$ set; an independent countable Dirichlet set which is not Kronecker; a collection of $q$-disjoint Kronecker sets whose union is independent but Helson $1/q$; A countable collection of disjoint Kronecker sets whose union is closed and independent but not Helson: a perfect independent Dirichlet set which is not Helson.

Let $H_n$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_n$. So $L^2\left(H_n\right)$ has a natural structure of G-module. We obtain a decomposition of $L^2\left(H_n\right)$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^2\left(H_n\right)$ where $L={\sum }_{j=1}^p(X_j^2+Y_j^2)-{\sum }_{j=p+1}^n(X_j^2+Y_j^2)$, and where $X_1,Y_1,...,X_n,Y_n,T$ denotes the standard basis of the Lie algebra of $H_n$. Finally, we obtain a spectral characterization of the...

We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.

Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^p\left(I\right)$, 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$ and their preduals.

Consider I = [0,1] as a compact topological semigroup with max multiplication and usual topology, and let $C\left(I\right),L^p\left(I\right),1\le p\le \infty$, be the associated algebras. The aim of this paper is to study the spaces $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$, r > p, and their preduals.

We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.

Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L_K^1\left(H_n\right)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L_K^1\left(H_n\right)$ can be identified with the set $\Delta (K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $\Delta (K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $\Delta (K,H_n)$ is a complete...