Necessary conditions and sufficient conditions of irregular shearlet frames |
| |
Authors: | Shenming Jiang Zetao Jiang Shaoqin Zhang |
| |
Institution: | 1. College of Computer Science and Technology, Nanjing Uinversity of Aeronautics and Astronautics, China;2. College of Mathematics and Information Science, Nanchang HangKong Universty, China;3. +86 4. 188 5. 7731 6. 9333;7. School of Computer Science and Engineering, Guilin University of Electronic Technology, China;8. 181 9. 6709 10. 7638 |
| |
Abstract: | 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. |
| |
Keywords: | Irregular shearlet system shearlet density relatively separated shearlet frame lower frame bound upper frame bound stability 94A12 42C15 |
|
|