Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

Abstract

We develop a distributional framework for the shearlet transform Sψ0.16em:0.16emS0true(R2true)0.16em→0.16emscriptS(S)${\mathcal {S}}_{\psi }\,{:}\, {\mathcal {S}}_0\big (\mathbb {R}^2\big )\,{\rightarrow }\,{\mathcal {S}}(\mathbb {S})$ and the shearlet synthesis operator Sψt-0.16em:-0.16emscriptS(S)-0.16em→-0.16emS0true(R2true)${\mathcal {S}}^t_{\psi }\!: \!{\mathcal {S}}(\mathbb {S})\!\rightarrow \!{\mathcal {S}}_0\big (\mathbb {R}^2\big )$, where S0true(R2true)${\mathcal {S}}_0\big (\mathbb {R}^2\big )$ is the Lizorkin test function space and scriptSfalse(double-struckSfalse)${\mathcal {S}}(\mathbb {S})$ is the space of highly localized test functions on the standard shearlet group S$\mathbb {S}$. These spaces and their duals S0′(double-struckR2),0.16emS′(S)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)},\, \mathcal {S}^\prime (\mathbb {S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector ψ belongs to S0true(R2true)${\mathcal {S}}_0\big (\mathbb {R}^2\big )$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from S0′(double-struckR2)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)}$ to S′(S)$\mathcal {S}^\prime (\mathbb {S})$ and from S′(S)$\mathcal {S}^\prime (\mathbb {S})$ to S0′true(R2true)$\mathcal {S}_0^\prime \big (\mathbb {R}^2\big )$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.