Bandlimited spaces on some 2-step nilpotent Lie groups with one Parseval frame generator

Abstract

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 dim P/Z = dim M . We study bandlimited subspaces of L 2 (N ) which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of bandlimited 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 Γ ⊂ N . Finally, we show some explicit examples in the last section.

Sampling and interpolation on some nilpotent Lie groups

“…The results in the proposition above are some facts, which are well-known in the theory of harmonic analysis of nilpotent Lie groups. See [8], where we specialized to the class of groups considered here. For general nilpotent Lie groups, we refer the interested reader to Section 4.3 in [2] which contains a complete presentation of the Plancherel theory of nilpotent Lie groups.…”

“…Moreover, sampling spaces using a similar definition of bandlimitation were studied in [8] and [7] for a class of nilpotent Lie groups which contains the Heisenberg Lie groups. This class of groups was first introduced by the author in [8]. However, nothing was said about the interpolation property of the sampling spaces described in [8].…”

“…This class of groups was first introduced by the author in [8]. However, nothing was said about the interpolation property of the sampling spaces described in [8]. In fact, the question of existence of sampling spaces with interpolation property on some noncommutative nilpotent Lie groups is a challenging problem which is the central focus of this paper.…”

Regular Sampling on Metabelian Nilpotent Lie Groups: The Multiplicity-Free Case

“…Firstly, we observe that if N = R d then Γ can be taken to be an integer lattice, and the Hilbert space of functions vanishing outside the cube - It is shown in [9,6] that there exist subspaces of L 2 (N) which are sampling subspaces with respect to Γ. We have also established in [19,18,17] the existence of sampling spaces defined over a class of simply connected, connected nilpotent Lie groups which satisfy the following conditions: N is a step-two nilpotent Lie group with Lie algebra n of dimension n such that n = a ⊕ b ⊕ c where [a, b] ⊆ c, a, b are commutative Lie algebras, a = R-span {X 1 , X 2 , · · · , X d } , b = R-span {Y 1 , Y 2 , · · · , Y d } , c = R-span {Z 1 , Z 2 , · · · , Z n-2d } (d ≥ 1, n > 2d) and…”

Sinc-type functions on a class of nilpotent Lie groups

“…H Ω = f ∈ L 2 (R) : support (F f ) ⊂ [−Ω, Ω] we have the following reconstruction formula (1) f (x) = n∈Z f πn Ω sin (π (x − k)) (π (x − k)) and we say that H Ω is a sampling subspace of L 2 (R) with respect to the lattice π Ω Z. A relatively novel problem in abstract harmonic analysis has been to find analogues of (1) for other locally compact groups [6,4,14,15,12,13]. Any attempt to generalize the given formula above leads to several obstructions.…”