Frames arising from irreducible solvable actions I

Oussa, Vignon

doi:10.1016/j.jfa.2017.07.009

Cited by 18 publications

(21 citation statements)

References 27 publications

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“…Moreover, the following establishes the existence of a Parseval frame generated by a continuous and compactly supported function when H is exponential and solvable. This does close a significant gap in our previous work [28]. A few additional observations are in order.…”

Section: Main Theoremssupporting

confidence: 84%

“…Precisely, they are obtained by inducing a unitary character of a normal closed subgroup. It turns out that for a surprisingly large class of solvable Lie groups, similar constructions are possible [28,20]. One may then ask the following: to which extent can we generalize the results in [28,20] in a unified manner?…”

Section: Introduction and Overview Of The Workmentioning

confidence: 99%

“…These types of frames naturally arise in the context of shearlet, time-frequency analyses, and wavelet theory [23,24,25,18,14,13,16] where frames generated by functions satisfying certain regular conditions (such as fast decay and various degrees of smoothness) are often sought after. As such, the present project, which was initiated in [28] provides a unified discretization scheme relevant to these topics. Furthermore, our main results (see Theorem 1 and Theorem 2) extend the work of [28] to a significantly broader class of representations (see Proposition 24 for an example).…”

Section: Introduction and Overview Of The Workmentioning

confidence: 99%

“…As such, the present project, which was initiated in [28] provides a unified discretization scheme relevant to these topics. Furthermore, our main results (see Theorem 1 and Theorem 2) extend the work of [28] to a significantly broader class of representations (see Proposition 24 for an example). While the central results of [28] only focus on cases where the conormal subgroup H of G is completely solvable, we allow H to be any connected Lie group here (see for example, Proposition 24.)…”

Section: Introduction and Overview Of The Workmentioning

confidence: 99%

“…On the other hand, the operators π (z) (where z ∈ H) act on H by left translations and the combined action of the normal and conormal parts of the group gives an action resembling that of time-frequency translations [3]. Perhaps, it is also worth highlighting that all irreducible representations of groups connected to Gabor theory, wavelet and shearlet analyses naturally arise in this fashion (see [28,Section 1.4]).…”

Section: Introduction and Overview Of The Workmentioning

confidence: 99%

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Compactly supported bounded frames on Lie groups

Oussa¹

2019

Journal of Functional Analysis

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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.

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Section: Main Theoremssupporting

confidence: 84%