A simple proof of the existence of sampling spaces with the interpolation property on the Heisenberg group

A surprisingly short geometric proof of the existence of sampling spaces with the interpolation property on the Heisenberg Lie group is given. This result was originally proved by of B. Currey and A. Mayeli [3]. In the loving memory of my father Dr. Germain Oussa. Let N be a locally compact group, and let Γ be a discrete subset of N. Let H be a leftinvariant closed subspace of L (N) consisting of continuous functions. We call H a sampling space (Section 2.6 [5]) with respect to Γ if the restriction mapping RΓ : H → l (Γ) , RΓf = (f (γ))γ∈Γ is an isometry, and there exists a vector s ∈ H such that for any vector f ∈ H, f (x) = ∑ γ∈Γ f (γ) s (γ −1x) with convergence in the norm of H. If RΓ is onto, then RΓ is a unitary map and we say that the sampling space H has the interpolation property. For example the vector space of square-integrable continuous functions on the real line whose Fourier transforms are supported on the interval [ − 2 , 1 2 ] is a sampling space with interpolation property with respect to the lattice subgroup Z. This example is provided by the well-known Whittaker, Shannon, Kotel’nikov Theorem (see Example 2.52 [5].) As far as I know, the first example of a sampling space with interpolation property on a non-commutative nilpotent Lie group using the Plancherel transform was defined over the three-dimensional Heisenberg Lie group. This remarkable example is due to Currey and Mayeli [3]. In [6], I gave sufficient conditions for the existence of sampling spaces with the interpolation property on a class of non-commutative nilpotent Lie groups. The work presented in [6, 3] suggests that in general the investigation of sampling spaces with the interpolation property over non-commutative groups is by no mean an easy task. The main objective of the present work is to offer a surprisingly simpler and shorter proof (than the one given in [3]) of the fact that there exist sampling subspaces defined over the Heisenberg group which also enjoy the interpolation property with respect to a discrete uniform subgroup. I would like to point out that the proof given here is not a substitute for the work of Currey and Mayeli in [3]. In fact, the authors of [3] prove more than the mere existence of a sampling space with the interpolation property over the Heisenberg group. They also gave an explicit construction of such a space.