Analysis vs. synthesis sparsity for \(\alpha\)-shearlets

• Published in: arXiv.org 2017-02
• Main Authors: Voigtlaender, Felix, Pein, Anne
• Format: Article
• Language: English
• Subjects: Approximation; Atomic properties; Decay; Economic models; Mathematical analysis; Smoothness; Sparsity; Synthesis
• ISSN: 2331-8422

There are two notions of sparsity associated to a frame \(\Psi=(\psi_i)_{i\in I}\): Analysis sparsity of \(f\) means that the analysis coefficients \((\langle f,\psi_i\rangle)_i\) are sparse, while synthesis sparsity means that \(f=\sum_i c_i\psi_i\) with sparse coefficients \((c_i)_i\). Here, sparsity of \(c=(c_i)_i\) means \(c\in\ell^p(I)\) for a given \(p0\). 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 \(\mathscr{S}_s^{p,q}\) introduced by Labate et al. Hence, membership in \(\mathscr{S}_s^{p,q}\) 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 \(\epsilon>0\), then \(\Vert f-f_N\Vert_{L^2}\lesssim N^{-(1-\epsilon)}\), where \(f_N\) 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 \(\tilde{\Psi}\) of \(\Psi\), not w.r.t. \(\Psi\) itself. This is not completely satisfying, since the properties of \(\tilde{\Psi}\) (decay, smoothness, etc.) are largely unknown. We also consider \(\alpha\)-shearlet systems. For these, the shearlet smoothness spaces have to be replaced by \(\alpha\)-shearlet smoothness spaces. We completely characterize the embeddings between these spaces, allowing us to decide whether sparsity w.r.t. \(\alpha_1\)-shearlets implies sparsity w.r.t. \(\alpha_2\)-shearlets.