Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896776 | Journal of Functional Analysis | 2018 | 53 Pages |
Abstract
In this work, we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable Lie groups. In contrast to other well-known techniques such as the coorbit theory, the generalized coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Moreover, our scheme produces explicit constructions of frames with precise frame bounds. As an illustration of the scope of our results, we highlight that a large class of representations which naturally occur in wavelet theory and time-frequency analysis is handled by our scheme. For example, the affine group, the generalized Heisenberg groups, the shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are a few examples of groups whose irreducible representations are handled by our method. The class of representations studied in this work is described as follows. Let G be a simply connected, connected, completely solvable Lie group with Lie algebra g=p+m. Next, let Ï be an infinite-dimensional unitary irreducible representation of G obtained by inducing a character from a closed normal subgroup P=expâ¡p of G. Additionally, we assume that G=PâM, M=expâ¡m is a closed subgroup of G, dμM is a fixed Haar measure on the solvable Lie group M and there exists a linear functional λâpâ such that the representation Ï=Ïλ=indPG(Ïλ) is realized as acting in L2(M,dμM). Making no assumption on the integrability of Ïλ, we describe explicitly a discrete subset Î of G and a vector fâL2(M,dμM) such that Ïλ(Î)f is a tight frame for L2(M,dμM). We also construct compactly supported smooth functions s and discrete subsets ÎâG such that Ïλ(Î)s is a frame for L2(M,dμM).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Vignon Oussa,