Bandlimited Spaces on Some 2-step Nilpotent Lie Groups with One Parseval Frame Generator
Let N be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let Z be the center of N. Assume that N = P ⋊M such that P, and M are simply connected, connected abelian Lie groups, M acts non-trivially on P by automorphisms and dimP/Z = dimM. We study band-limited subspaces of L 2 (N) which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of band-limited subspaces of L 2 (N) which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set ⊂ N. Finally, we show some explicit examples in the last section.