Democracy of shearlet frames with applications

Abstract

Shearlets on the cone provide Parseval frames for L2. They also provide near-optimal approximation for the class E of cartoon-like images. Moreover, there are spaces associated to them other than L2 and there exist embeddings between these and classical spaces.We prove approximation properties of the cone-adapted shearlet system coefficients in a more general context. Namely, when the target shearlet sequence belongs to a class or space different to that obtained from a shearlet sequence of a f∈E and when the error is not necessarily measured in the L2-norm (or, since the shearlet system is a frame, the ℓ2-norm) but in a norm of a much wider family of smoothness spaces of “high” anisotropy. We first prove democracy of shearlet frames in shear anisotropic inhomogeneous Besov and Triebel–Lizorkin sequence spaces. Then, we prove embeddings between approximation spaces and discrete weighted Lorentz spaces in the framework of shearlet coefficients. Simultaneously, we also prove that these embeddings are equivalent to Jackson and Bernstein type inequalities. This allows us to find real interpolation between these highly anisotropic sequence spaces. We also describe how some of these results can be extended to other shearlet and curvelet generated spaces. Finally, we show some examples of embeddings between wavelet approximation spaces and shearlet approximation spaces and obtain a similar result stated in L2(R2) for the curvelet smoothness spaces. This also paves the way to the use of thresholding algorithms in compression or noise reduction.