Multi-window Gabor frames and oblique Gabor duals on discrete periodic sets  

作　　者：LI YunZhang LIAN QiaoFang 

机构地区：[1]College of Applied Sciences, Beijing University of Technology, Beijing 100124, China [2]Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

出　　处：《Science China Mathematics》2011年第5期987-1010,共24页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 10901013, 10671008);Beijing Natural Science Foundation (Grant No. 1092001);the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005030);the Project Sponsored by SRF for ROCS, SEM of China

摘　　要：Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l2(S), and show that such condition is suficient for theexistence of g={XE1:0≤l≤L-1} with G(g,N,m) We characterize g with G(g,N,m) being respectively a frame for L(g,N,m) being a tight frame for l2(S).and G(h, N, M ) in L(h, N, M ), we establish a criterion for the existence of an oblique Gabor dual of g in L(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2（S） be the closed subspace of l2（Z） consisting of sequences vanishing outside S. For f = { fl ： 0≤l≤L-1 }l2（Z）, we denote by G（f, N, M） the Gabor system generated by f, and by L（f, N, M） the closed linear subspace generated by G（f, N, M）. This paper addresses density results, frame conditions for a Gabor system G（g, N, M） in l2（S）, and Gabor duals of the form G（a, N, M） in some L（h, N, M） for a frame G（g, N, M） in l2（S） （so-called oblique duals）. We obtain a density theorem for a Gabor system G（g, N, M） in l2（S）, and show that such condition is suficient for theexistence of g={XE1：0≤l≤L-1} with G（g,N,m） We characterize g with G（g,N,m） being respectively a frame for L（g,N,m） being a tight frame for l2（S）.and G（h, N, M ） in L（h, N, M ）, we establish a criterion for the existence of an oblique Gabor dual of g in L（h, N, M）, study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals （among which the one with the smallest norm is obtained）. Some examples are also provided.

关 键 词：Gabor system Gabor flame multi-window Gabor frame periodic set oblique Gabor dual 

分 类 号：O314[理学—一般力学与力学基础] TP391.41[理学—力学]