Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

Frames generated by compact group actions

“…Bownik [10] seems to have the first results along these lines. His example was followed in [9,12,13,17,34,39]. We now carry that tradition to the setting of compact, nonabelian subgroups.…”

“…Their work was released at approximately the same time, and each cites the other, so it is not clear who deserves credit for this line of research. The idea of applying a Fourier-like transform and classifying invariant subspaces in terms of range functions has since been applied by a host of researchers in a variety of settings [1,3,4,10,11,12,13,14,17,19,30,39]. Recently, the author [34] and Hernández, et al [9] independently applied a version of the Zak transform to classify translation invariance by an abelian subgroup.…”

“…We emphasize the novelty of applying this technique with a nonabelian subgroup. Of the results mentioned above, only Currey, et al [17] treats a noncommutative case. 1 The theory of translation invariance in the nonabelian setting is only in its beginning stages.…”

“…Despite significant interest in the problem, very little has been done in the area of group frames with multiple generators. The most fruitful area has been frames generated by translations, mostly with abelian groups [9,10,12,13,34,39] but in at least one case with nonabelian [17]. In the setting of discrete nonabelian groups, Hernández and his collaborators [8] have recently developed an abstract machinery to handle frames with multiple generators for a special class of unitary representations.…”

Left translates of a square integrable function on the Heisenberg group

“…In [1], the authors introduced bracket map on the polarized Heisenberg group H n pol using the group Fourier transform and obtained characterizations of orthonormal system, frames and Riesz basis consisting of left translates of ϕ in L 2 (H n pol ) in terms of the bracket map. In [6], Currey et al generalized some results of [3] to shift-invariant spaces associated with a class of nilpotent Lie groups. The concept of the bracket map has been generalized in [2] to include any nonabelian discrete group Γ using its unitary representations and L 1 space over the non-commutative measurable space vNa(Γ), which is the compact dual of Γ whose underlying space is a group von Neumann algebra.…”