Sinc-type functions on a class of nilpotent Lie groups

Abstract

Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra 𝔫 is an n-dimensional vector space over the reals. Moreover, 𝔫=𝔷⊕𝔟⊕𝔞, 𝔷 is the center of 𝔫, 𝔷=ℝZn-2d⊕ℝZn-2d-1⊕⋯⊕ℝZ1, 𝔟=ℝYd⊕ℝYd-1⊕⋯⊕ℝY1, 𝔞=ℝXd⊕ℝXd-1⊕⋯⊕ℝX1. Next, assume 𝔷⊕𝔟 is a maximal commutative ideal of 𝔫, [𝔞,𝔟]⊆𝔷, and det ([Xi,Yj])1≤i,j≤d is a non-trivial homogeneous polynomial defined over the ideal [𝔫,𝔫]⊆𝔷. We do not assume that [𝔞,𝔞] is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of L2(N) with respect to some discrete set Γ. The set Γ is explicitly constructed by fixing a strong Malcev basis for 𝔫. We provide sufficient conditions for which a function f is determined from its sampled values on (f(γ))γ∈Γ. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.