Sampling Theorems for Some Two-Step Nilpotent Lie Groups

الأصل · EN

Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie Lie algebra n is an n-dimensional vector space over the reals. Moreover, n=z, z is the center of n, z =RZₙ₋₂dₙ₋₂d₋₁⊕⊕ RZ₁, b =RYd Yd₋₁⊕₁, a =RXd₋₁⊕⊕ RX₁. Next, assume z is a maximal commutative ideal of n, [a,b], and det([Xᵢ,Yⱼ])₁≤ ᵢ,ⱼ≤ d is a non-trivial homogeneous polynomial defined over the ideal [n,n]. 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 Jordan-Hölder 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.