切波框架的存在充分条件及其稳定性  

On sufficient conditions and stability of shearlet frame

作　　者：江慎铭[1,2] 江泽涛[3] 

机构地区：[1]南京航空航天大学计算机科学与技术学院,南京210016 [2]南昌航空大学数信学院,南昌330063 [3]桂林电子科技大学计算机科学与工程学院,广西541004

出　　处：《中国科学：数学》2014年第8期899-928,共30页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:61272216和61163048);江西省教育厅科技基金(批准号:GJJ11513);航空创新基金(批准号:CASC201102);江西省科学技术合作(批准号:2010EAB01700);江西自然科学基金(批准号:20114BAB201035)资助项目

摘　　要：2011年,Kittipoom等人引入了一类新的切波生成函数空间,并指出此空间拥有许多优秀的性质,例如,该空间在平方可积函数空间中稠密,由该空间中元素生成的切波框架拥有强齐次逼近性质等.本文的主要目的是研究由Kittipoom等人引入的切波生成函数空间中的元素生成切波框架的充分条件及由该空间中的元素生成的切波框架的稳定性.具体而言,首先参考由Dahlke等人引入的切波群的定义将Kittipoom等人引入的切波群的定义进行适当调整,使得由Kittipoom等人引入的切波生成函数空间中每个元素都是可允许的;其次得到由该切波生成函数空间中任意一个元素和任意一个相对分离的稠密点列可形成一个切波框架;最后证明这些框架在时间、尺度和剪切参数或生成函数发生小扰动时仍然形成切波框架.这些结论使得切波框架在工程应用方面有着极大的灵活性和实用性.Abstract In 2011, Kittipoom et al. introduced a new class of shearlet generators, and gave many good properties. For example, the new class of shearlet generators is dense in the space of square integrable functions and the shearlet frame generated by a member of the new class of shearlet generators possesses the strong HAP etc. In this paper, our main goal is to study sufficient conditions and the stability of shearlet frame. Specifically, we first adjust the definition of shearlet group introduced by Kittipoom et al. through referring to the definition of shearlet group introduced by Dahlke et al., and make each element of the new class of shearlet generators to be admissible. Secondly, we show that a function from the new class of shearlet generators generates shearlet frames for every sufficiently dense and relatively separated sequence. Finally, we prove that a frame generated by a function from the space remains a frame when the time, scale and shear parameters and the generating function undergo small perturbations. These conclusions make shearlet framework to possess great flexibility and practicality in engineering applications.

关 键 词：切波群 切波 切波变换 切波框架 

分 类 号：O174.2[理学—数学]