Nonlinear approximation in <i>α</i> ‐modulation spaces

Borup, Lasse; Nielsen, Morten

doi:10.1002/mana.200310348

Cited by 31 publications

(32 citation statements)

References 24 publications

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“…Notice that the dilation matrices A (1) , A (2) are associated with anisotropic dilations and, more specifically, parabolic scaling dilations; by contrast, the shear matrices B (1) , B (2) are non-expanding and their integer powers control the directional features of the shearlet system. Hence, the systems (3.11) form collections of well-localized functions defined at various scales, orientations and locations, controlled by the indices j, ℓ, k respectively.…”

Section: The Shearlet Representationmentioning

confidence: 99%

“…• the boundary shearlets { ψ j,ℓ,k : j ≥ 0, ℓ = ±2 j , k ∈ Z 2 }, obtained by joining together slightly modified versions of ψ (1) j,ℓ,k and ψ (2) j,ℓ,k , for ℓ = ±2 j , after that they have been restricted in the Fourier domain to the cones P 1 and P 2 , respectively. The precise definition is given below.…”

Section: The Shearlet Representationmentioning

confidence: 99%

“…In that case, we will indicate the matrices A (1) and B (1) with A and B, and the shearlets in P 1 simply with ψ j,ℓ,k .…”

Section: The Shearlet Representationmentioning

confidence: 99%

“…Let {ϕ j,ℓ } be a BAPU corresponding to the shearlet-type covering T M given in Proposition 4.1. By the properties of the shearlet-like covering, it is clear that we can choose Ω such that supp (ϕ j,ℓ ) ⊂ supp (ω j ), ℓ ∈ L j , and supp (ω j ) ⊂ ∪ ℓ∈Lj supp (φ j,l ), 1 Here the Q-moderate function w is the first axis projection, and the sequence of points x i ∈ Q i is given by x i = (2 j , 0).…”

Section: Proposition 43 For 0 < P ≤ ∞ 0 < Q < ∞ and β ∈ R We Havementioning

confidence: 99%

“…For example, wavelets are 'naturally' associated with Besov spaces, and the notion of sparseness in the wavelet expansion is equivalent to an appropriate smoothness measure in Besov spaces [25]. Similarly, the Gabor systems, which are widely used in time-frequency analysis, are naturally associated to the class of modulation spaces [1,16]. In the case of shearlets, a sequence of papers by Dahlke, Kutyniok, Steidl and Teschke [8,9,10] have recently introduced a class of shearlet spaces within the framework of the coorbit space theory of Feichtinger and Gröchenig [14,15].…”

Section: Introductionmentioning

confidence: 99%

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Shearlet Smoothness Spaces

Labate

Mantovani

Negi

2013

J Fourier Anal Appl

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The shearlet representation has gained increasingly more prominence in recent years as a flexible mathematical framework which enables the efficient analysis of anisotropic phenomena by combining multiscale analysis with the ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.

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