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Necessary conditions and sufficient conditions of irregular shearlet frames
Author(s) -
Jiang Shenming,
Jiang Zetao,
Zhang Shaoqin
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400403
Subject(s) - shearlet , mathematics , frame (networking) , scale (ratio) , space (punctuation) , pure mathematics , computer science , image (mathematics) , artificial intelligence , telecommunications , physics , quantum mechanics , operating system
In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space‐scale‐shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space‐scale‐shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space‐scale‐shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo‐spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.