Coorbit Spaces with Voice in a Fréchet Space

Dahlke, Stephan; Mari, Filippo De; Vito, Ernesto De; Labate, Demetrio; Steidl, Gabriele; Teschke, Gerd; Vigogna, Stefano

doi:10.1007/s00041-016-9466-x

Cited by 16 publications

(35 citation statements)

References 46 publications

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“…We recall some basic properties of T w ; for proofs we refer to Theorem 4.3 of [5], which is based on results in [7]. We endow T w with the (unique) topology such that a sequence ( f n ) ∈N in T w converges to 0 if and only if lim n→+∞ f n L p, w = 0 for all 1 < p < ∞.…”

Section: An Overviewmentioning

confidence: 99%

“…Remark 2.3. i) Since w(x) ≥ 1, Assumption (5) implies that K ∈ L p (G) for all p > 1. If π is irreducible, this last fact gives that V is an isometry up to a constant, so that π is always a reproducing representation.…”

Section: An Overviewmentioning

confidence: 99%

“…If π is irreducible, this last fact gives that V is an isometry up to a constant, so that π is always a reproducing representation. If π is reducible, condition (5) is not sufficient to ensure that π is reproducing; however if K * K = K, then π is always reproducing. ii) If w −1 belongs to L q (G) for some 1 < q < ∞, then Hölder's inequality shows K ∈ L 1 (G), but in general K L 1,w (G).…”

Section: An Overviewmentioning

confidence: 99%

“…Theorem 2.6 (Theorem 4.4 of [5]). Under assumption (5), the following hold: i) for every Φ ∈ U w there exists a unique element π(Φ)u ∈ S ′ w such that…”

Section: An Overviewmentioning

confidence: 99%

See 3 more Smart Citations

On the Atomic Decomposition of Coorbit Spaces with Non-integrable Kernel

Dahlke

Mari

Vito

et al. 2019

Landscapes of Time-Frequency Analysis

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This paper ist concerned with recent progress in the context of coorbit space theory. Based on a square integrable group representation, the coorbit theory provides new families of associated smoothness spaces, where the smoothness of a function is measured by the decay of the associated voice transform. Moreover, by discretizing the representation, atomic decompositions and Banach frames can be constructed. Usually, the whole machinery works well if the associated reproducing kernel is integrable with respect to a weighted Haar measure on the group. In recent studies, it has turned out that to some extent coorbit spaces can still be established if this condition is violated. In this paper, we clarify in which sense atomic decompositions and Banach frames for these generalized coorbit spaces can be obtained.2010 Mathematics Subject Classification. 46E35, 43A15, 42B35, 22D10, 42B15.

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Section: An Overviewmentioning

confidence: 99%

Section: An Overviewmentioning

confidence: 99%

Section: An Overviewmentioning

confidence: 99%

“…Theorem 2.6 (Theorem 4.4 of [5]). Under assumption (5), the following hold: i) for every Φ ∈ U w there exists a unique element π(Φ)u ∈ S ′ w such that…”

Section: An Overviewmentioning

confidence: 99%

See 2 more Smart Citations

On the Atomic Decomposition of Coorbit Spaces with Non-integrable Kernel

Dahlke

Mari

Vito

et al. 2019

Landscapes of Time-Frequency Analysis

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“…Their theory hinges on the existence of a cyclic vector satisfying a reproducing formula equation together with other technical assumptions. Other generalizations of coorbit theory can be found in [6,12,30].…”

Section: Introduction and Overview Of The Workmentioning

confidence: 99%

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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General Coorbit Space Theory for Quasi-Banach Spaces and Inhomogeneous Function Spaces with Variable Smoothness and Integrability

Kempka

Schäfer

Ullrich³

2016

J Fourier Anal Appl

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In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gröchenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich [50] and Rauhut [48] to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov-Lizorkin-Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.

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Coorbit Spaces with Voice in a Fréchet Space

Cited by 16 publications

References 46 publications

On the Atomic Decomposition of Coorbit Spaces with Non-integrable Kernel

On the Atomic Decomposition of Coorbit Spaces with Non-integrable Kernel

Compactly supported bounded frames on Lie groups

General Coorbit Space Theory for Quasi-Banach Spaces and Inhomogeneous Function Spaces with Variable Smoothness and Integrability

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