In the first part of the paper a general notion of sampling expansions for locally compact groups is introduced, and its close relationship to the discretisation problem for generalised wavelet transforms is established. In the second part, attention is focussed on the simply connected nilpotent Heisenberg group H. We derive criteria for the existence of discretisations and sampling expansions ...

The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover , the Hardy-Littlewood-Sobolev inequality is established.

A mathematical construction of the conformal field theory (CFT) associated to a compact torus, also called the “nonlinear σ-model” or “lattice-CFT”, is given. Underlying this approach to CFT is a unitary modular functor, the construction of which follows from a “Quantization commutes with reduction”type of theorem for unitary quantizations of the moduli spaces of holomorphic torusbundles and ac...

In this article, we introduce a kind of unitary operator U associated with the involution on the Heisenberg group, invariant closed subspaces are identified with the characterization spaces of sub-Laplacian operators. In the sense of vector-valued functions, we study the theory of continuous wavelet transform. Also, we obtain a new inversion formula of Radon transform on the Heisenberg group Hn.

We show that the quantum Heisenberg group Hq(1) can be obtained by means of contraction from quantum SUq(2) group. Its dual Hopf algebra is the quantum Heisenberg algebra Uq(h(1)). We derive left and right regular representations for Uq(h(1)) as acting on its dual Hq(1). Imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representati...

In this note I study the Sasakian geometry associated to the standard CR structure on the Heisenberg group, and prove that the Sasaki cone coincides with the set of extremal Sasakian structures. Moreover, the scalar curvature of these extremal metrics is constant if and only if the metric has Φsectional curvature −3. I also briefly discuss some relations with the well-know sub-Riemannian geomet...

This note describes some computational results that were obtained while studying the associated group of automorphisms Aut(H), [2]. Specifically, we give an explicit description of a splitting of this group as an extension of GL(2,Z) by Z. That such a splitting exists could, in principle, be shown by computing a 2-cocycle for the extension and showing that it represents 0 in the cohomology grou...

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.