Examples of Coorbit Spaces for Dual Pairs

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.

Section: Condition 17mentioning

confidence: 99%

Section: Introduction and Overview Of The Workmentioning

confidence: 99%

Compactly supported bounded frames on Lie groups

Oussa¹

2019

“…The samplability is one of most important topics in sampling theory, see for instance [22,26,46] for band-limited signals, [4,43] for signals in a shift-invariant space, [16,20,21,24,25] for signals in a co-orbit space, and [27,33] for signals in reproducing kernel Hilbert and Banach spaces. In this paper, we study the samplability of signals in a reproducing kernel subspace of L p (R d ) associated with an idempotent operator.…”

Section: And A4 In Appendixmentioning

confidence: 99%

“…(A.13)ThenT δ 0 T = T T δ 0 = T δ (x, y) − K(x, y) R d K(x, z) ω δ 0 (K)(z, y) dz (A.15)by Theorem A.1. ThereforeT δ 0 f − Tf p r 1 (δ 0 ) f p for all f ∈ L p , (A 16). …”

mentioning

confidence: 98%

Sampling and reconstruction of signals in a reproducing kernel subspace of Lp(Rd)

Nashed

Sun

2010

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of L p (R d ), 1 p ∞, associated with an idempotent integral operator whose kernel has certain offdiagonal decay and regularity. The space of p-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of L p (R d ). We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation-projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap.

“…More recently, the introduction of shearlets (at least the group-theoretic version) triggered the systematic study of the associated coorbit spaces [5,6]. Coorbit spaces over the Blaschke group and their connection to complex analysis are discussed in [13]; see also [4]. The recent papers [18,19] pointed out that the study of wavelet coorbit spaces could be considerably extended to cover a multitude of group-theoretically defined wavelet systems in a unified approach that allows to prove the existence of easily constructed, nice wavelet systems and atomic decompositions in a large variety of settings.…”

Section: Introductionmentioning

confidence: 99%

Wavelet coorbit spaces viewed as decomposition spaces

Führ

Voigtlaender

2015

In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces L p,q v with respect to the quasi-regular representation of a semi-direct product R d H with suitably chosen dilation group H, and certain decomposition spaces D (Q, L p , q u ) (essentially as introduced by Feichtinger and Gröbner) where the localized "parts" of a function are measured in the F L p -norm. This equivalence is useful in several ways: It provides access to a Fourier-analytic understanding of wavelet coorbit spaces, and it allows to discuss coorbit spaces associated to different dilation groups in a common framework. As an illustration of these points, we include a short discussion of dilation invariance properties of coorbit spaces associated to different types of dilation groups.