Analysis vs. synthesis sparsity for α-shearlets

Kurzfassung/Abstract

There are two notions of sparsity associated to a frame Ψ=(ψi)i∈I: Analysis sparsity of f means that the analysis coefficients (⟨f,ψi⟩)i are sparse, while synthesis sparsity means that f=∑iciψi with sparse coefficients (ci)i. Here, sparsity of c=(ci)i means c∈ℓp(I) for a given p<2. We show that both notions of sparsity coincide if Ψ=SH(φ,ψ;δ) is a discrete (cone-adapted) shearlet frame with 'nice' generators φ,ψ and fine enough sampling density δ>0. The required 'niceness' is explicitly quantified in terms of Fourier-decay and vanishing moment conditions.
Precisely, we show that suitable shearlet systems simultaneously provide Banach frames and atomic decompositions for the shearlet smoothness spaces Sp,qs introduced by Labate et al. Hence, membership in Sp,qs is simultaneously equivalent to analysis sparsity and to synthesis sparsity w.r.t. the shearlet frame.
As an application, we prove that shearlets yield (almost) optimal approximation rates for cartoon-like functions f: If ϵ>0, then ∥f−fN∥L2≲N−(1−ϵ), where fN is a linear combination of N shearlets. This might appear to be well-known, but the existing proofs only establish this approximation rate w.r.t. the dual Ψ~ of Ψ, not w.r.t. Ψ itself. This is not completely satisfying, since the properties of Ψ~ (decay, smoothness, etc.) are largely unknown.
We also consider α-shearlet systems. For these, the shearlet smoothness spaces have to be replaced by α-shearlet smoothness spaces. We completely characterize the embeddings between these spaces, allowing us to decide whether sparsity w.r.t. α1-shearlets implies sparsity w.r.t. α2-shearlets.