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The Shearlet Transform and Lizorkin Spaces

by F. Bartolucci and S. Pilipovic and N. Teofanov

(Report number 2020-44)

Abstract
We prove a continuity result for the shearlet transform when restricted to the space of smooth and rapidly decreasing functions with all vanishing moments. We define the dual shearlet transform, called here the shearlet synthesis operator, and we prove its continuity on the space of smooth and rapidly decreasing functions over \(\mathbb{R}^2\times\mathbb{R}\times\mathbb{R}^{\times}\). Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions, and we prove its consistency with the classical definition for test functions. Our proofs are based on the strict relation between the shearlet, the wavelet and the affine Radon transforms proved in an earlier paper ([ACHA, 2019]) by one of the authors.

Keywords: shearlet transform; wavelet transform; Radon transform; Ridgelet transform; Lizorkin spaces

@Techreport{BPT20_917,
  author = {F. Bartolucci and S. Pilipovic and N. Teofanov},
  title = {The Shearlet Transform and Lizorkin Spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-44},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-44.pdf },
  year = {2020}
}

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