Frame Decomposition of Decomposition Spaces

Abstract. We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. Furthermore, we show embedding relations of traces of these subspaces with respect to the real axes.

“…Embedding results in Besov spaces have also been shown for the curvelet setting by Borup and Nielsen [4]. However, the technique used by these authors is completely different.…”

Section: 3mentioning

confidence: 90%

Shearlet Coorbit Spaces: Compactly Supported Analyzing Shearlets, Traces and Embeddings

Dahlke

Steidl

Teschke

2011

“…Unlike the theory of shearlet coorbit spaces, the approach presented here does not require any group structure and is closely associated with the geometrical properties of the spatial-frequency decomposition of the shearlet construction. Our method is derived from the theory of decomposition spaces originally introduced by Feichtinger and Gröbner [12,13] and recently revisited by Borup and Nielsen [2], who have adapted the theory of decomposition spaces to design a very elegant framework for the construction of smoothness spaces closely associated with particular structured decompositions in the Fourier domain. As will be made clear below, this approach can be viewed as a refinement of the classical construction of Besov spaces, which are associated with the dyadic decomposition of the Fourier space.…”

Section: Introductionmentioning

confidence: 99%

Shearlet Smoothness Spaces

Labate

Mantovani

Negi

2013

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.

“…multimodal spectral measurements in meteorology. To our best knowledge, it seems that there exist only few results in this direction: some important progress has been achieved for the curvelet case in [1] and for surfacelets in [16]. However, for the shearlet approach the question has been completely open.…”

Section: Introductionmentioning

confidence: 99%

The Continuous Shearlet Transform in Arbitrary Space Dimensions

Dahlke

Steidl

Teschke

2009

108

154

Abstract:This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.