Compactly supported bounded frames on Lie groups

Abstract

Let G = N H be a Lie group where N, H are closed connected subgroups of G, and N is an exponential solvable Lie group which is normal in G. Suppose furthermore that N admits a unitary character χ λ corresponding to a linear functional λ of its Lie algebra. We assume that the map h → Ad h −1 * λ defines an immersion at the identity of H. Fixing a Haar measure on H, we consider the unitary representation π of G obtained by inducing χ λ . This representation which is realized as acting in L 2 (H, dµ H ) is generally not irreducible, and we do not assume that it satisfies any integrability condition. One of our main results establishes the existence of a countable set Γ ⊂ G and a function f ∈ L 2 (H, dµ H ) which is compactly supported and bounded such that {π (γ) f : γ ∈ Γ} is a frame. Additionally, we prove that f can be constructed to be continuous. In fact, f can be taken to be as smooth as desired. Our findings extend the work started in [28] to the more general case where H is any connected Lie group. We also solve a problem left open in [28]. Precisely, we prove that in the case where H is an exponential solvable group, there exist a continuous (or smooth) function f and a countable set Γ such that {π (γ) f : γ ∈ Γ} is a Parseval frame. Since the concept of well-localized frames is central to time-frequency analysis, wavelet, shearlet and generalized shearlet theories, our results are relevant to these topics and our approach leads to new constructions which bear potential for applications. Moreover, our work sets itself apart from other discretization schemes in many ways. (1) We give an explicit construction of Hilbert frames and Parseval frames generated by bounded and compactly supported windows. (2) We provide a systematic method that can be exploited to compute frame bounds for our constructions. (3) We make no assumption on the irreducibility or integrability of the representations of interest.

On dual molecules and convolution-dominated operators

“…In the absence of off-diagonal decay, even the existence of discrete expansions, without claims on the quality of the dual systems, is non-trivial, and was recently established in [45,72]; see also [10]. In the same spirit, the existence of discrete frames in the orbit of a possibly non-square-integrable representation (in particular, non-integrable representation) of a solvable Lie group has recently been proved in [74,75], by means of a specific construction.…”

Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…It should be mentioned that (non-localized) orthonormal bases in the orbit of a nilpotent Lie group could still exist by Theorem 1.2, and even for nilpotent Lie groups not admitting a lattice, cf. [53,90].…”

Balian-Low type theorems on homogeneous groups

“…and study their relation with the density of the index set Λ ⊆ G. The existence of frames and Riesz sequences of the form (1) for "sufficiently dense" respectively "sufficiently Balian-Low theorem asserts that if g ∈ L 2 (R) is a phase-space localized function, then the family of functions {e 2πil• g(• + k) : k, l ∈ Z} does not form a Riesz basis for L 2 (R); see [5,9]. In view of recent constructions of orthonormal bases in the orbit of unitary representations of nilpotent Lie groups [27] and solvable semi-direct products [36,37], it is of interest whether a Balian-Low type theorem holds for other groups than the Heisenberg group. In this regard we recall the famous Kirillov lemma asserting that any nilpotent Lie group admits a subgroup isomorphic to the Heisenberg group.…”