Applications Of Shearlets To Directional Regularity And Thresholding Estimation

Shearlet method is one of new multiscale representation methods based onwavelet analysis. It overcomes the limitation of traditional wavelets in dealingwith multidimensional data, because it not only possess much higher directionalsensitivity, but also provides essentially optimal sparse approximation for2-Dfunctions containing curve edge. This paper studies directional regularity of afunction, as well as approximation solutions of a class of linear inverse problemsby discrete shearlet systems.First, we construct a family of shearlets with analytic formulae, in order tobetter understand two applications of shearlets; study decaying and vanishingmoment properties of shearlets; prove the equivalence of two definitions for twotypes of H¨older spaces. All these results will be used in this paper.Second, we characterize the directional regularity of a function by shearlets.It turns out that if a function is smoother in the direction of a singular line than inother direction, then the discrete shearlet coefcients decay slower in the normaldirection of the singular line than that away from it. The converse still holds insome sense. Our results extend the work of Lakhonchai, Sumetkijakan and Sampo.Finally, we apply shearlet thresholding approach to the model Y=Kf+εWwith (K~+Kf)∧(ξ)=(b+|ξ|~2) α f (ξ). More precisely, we verify that three clas-sical operators (Radon, Bessel and Riesz fractional integration transform) satisfythis above condition; Based on shearlet expansion and thresholding method, weconstruct the shearlet thresholding estimator, and derive the convergence rate of estimators over the space ε~2(A); It turns out that our estimator attains theoptimal convergence rate in L~2sense when K takes either Bessel or Riesz frac-tional integration transform. Our results generalize Colonna-Easley-Guo-Labate’stheorem.