2012/03/22 by Daniel Vera, Vera, Daniel · 1 citation
Engineering · Mathematics · #42B25 #42B35 #42C40 #Advanced Harmonic Analysis Research #Advanced Numerical Analysis Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #math.FA #msc:42B25 #msc:42B35 #msc:42C40
paper · pdf · doi:10.48550/arxiv.1203.5136
30 pages
arxiv created 2012/03/22 · openalex publication_date 2012/03/22 · arxiv updated 2012/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The shearlets are a special case of the wavelets with composite dilation that, among other things, have a basis-like structure and multi resolution analysis properties. These relatively new representation systems have encountered wide range of applications, generally surpassing the performance of their ancestors due to their directional sensitivity. However, little is known about their relation with spaces other than L2. Here, we find a characterization of a kind of anisotropic inhomogeneous Triebel-Lizorkin spaces (to be defined) with the so called "shearlets on the cone" coefficients. We first prove the boundedness of the analysis and synthesis operators with the "traditional" shearlets coefficients. Then, with the development of the smooth Parseval frames of shearlets of Guo and Labate we are able to prove a reproducing identity, which was previously possible only for the L2 case. We also find some embeddings of the (classical) dyadic spaces into these highly anisotropic spaces, and viceversa, for certain ranges of parameters. In order to keep a concise document we develop our results in the "weightless" case (w=1) and give hints on how to develop the weighted case.