Shearlet Expansion Theory on Lizorkin-Type Spaces

Abstract

We develop a shearlet expansion theory for the Lizorkin-type spaces \(\mathcal{S}_{0}(\mathbb{R}^2)\) and \(\mathcal{S}'_{0}(\mathbb{R}^2)\). We prove that the shearlet series expansion with respect to a Parseval shearlet converges in the topology of these spaces and provide a topological characterization of the Lizorkin space of distributions in terms of shearlet coefficients. Finally, we apply our distributional shearlet expansion theory to analyze asymptotic properties of distributions and obtain several Tauberian-type results that characterize the quasiasymptotics and quasiasymptotically boundedness of Lizorkin distributions via the asymptotic behavior of their shearlet coefficients.