广义框架的不相交性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (3): 589-596.

广义框架的不相交性

张伟^1,*(),李云章²()

¹ 河南财经政法大学数学与信息科学学院郑州 450046
² 北京工业大学应用数理学院北京 100124

收稿日期:2019-05-24 出版日期:2020-06-26 发布日期:2020-07-15
通讯作者: 张伟 E-mail:zwfylhappy@126.com;yzlee@bjut.edu.cn
作者简介:李云章, E-mail:yzlee@bjut.edu.cn
基金资助:
国家自然科学基金(11761079);河南省高等学校重点科研项目(20A110013)

Disjointness of Generalized Frames

Wei Zhang^1,*(),Yunzhang Li²()

¹ School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou 450046
² College of Applied Sciences, Beijing University of Technology, Beijing 100124

Received:2019-05-24 Online:2020-06-26 Published:2020-07-15
Contact: Wei Zhang E-mail:zwfylhappy@126.com;yzlee@bjut.edu.cn
Supported by:
the NSFC(11761079);the MPHE of Henan Province(20A110013)

摘要/Abstract

摘要：

Hilbert空间中框架的不相交性是由Han和Larson首先提出的，它与超框架有着密切的联系，在超框架及框架的构造中扮演重要的角色.广义框架是通常框架的推广.该文利用超广义框架刻画了广义框架的不相交性、强不相交性及弱不相交性；借助所得结果，给出了已知定理的不同证明方法；建立了对偶广义框架之间强不相交相和弱不相交性之间的关系；最后利用给定的强不相交的广义框架构造新的（超）对偶广义框架，所得结论恢复已有结果.

关键词: 广义框架, 对偶广义框架, 不相交性

Abstract:

The notion of disjointness of frames in Hilbert spaces was firstly introduced by Han and Larson, it is closely related with super frames in Hilbert spaces, and plays an important role in construction of super frames and frames. G-frames is a generalization of frames in Hilbert spaces. In this paper, we establish characterization of disjointness of g-frames, strong disjointness of g-frames and weak disjointness of g-frames in terms of super g-frames; With the results obtained, we give different proof method of the known theorem; We obtain the relation between strongly disjoint and weakly disjoint of dual g-frames; Finally, we use strong disjointness of g-frames to construct (super) dual g-frames, which cover the results obtained by other authors.

Key words: Generalized frames, Dual generalized frame, Disjointness

中图分类号:

O174.2

张伟,李云章. 广义框架的不相交性[J]. 数学物理学报, 2020, 40(3): 589-596.

Wei Zhang,Yunzhang Li. Disjointness of Generalized Frames[J]. Acta mathematica scientia,Series A, 2020, 40(3): 589-596.

参考文献 17

1	Duffin R J , Schaeffer A C . A class of nonharmonic Fourier series. Trans Amer Math Soc, 1952, 72, 341- 366 doi: 10.1090/S0002-9947-1952-0047179-6
2	Daubechies I , Grossmann A , Meyer Y . Painless nonorthogonal expansions. J Math Phy, 1986, 27, 1271- 1283 doi: 10.1063/1.527388
3	Chan R H , Riemenschneider S D , Shen L , et al. Tight frame:An efficient way for high-resolution image reconstruction. Appl Compute Harmon Anal, 2004, 17, 91- 115 doi: 10.1016/j.acha.2004.02.003
4	Strohmer T . Approximation of dual Gabor frames, window decay, and wireless communication. Appl Compute Harmon Anal, 2001, 11, 243- 262 doi: 10.1006/acha.2001.0357
5	Candès E J . Harmonic analysis of neural networks. Appl Compute Harmon Anal, 1999, 6, 197- 218 doi: 10.1006/acha.1998.0248
6	Eldar Y , Forney Jr G D . Optimal tight frames and quantum measurement. IEEE Trans Inform Theory, 2002, 48, 599- 610 doi: 10.1109/18.985949
7	Christensen O . An Introduction to Frames and Riesz Bases. Boston: Birkhäuser, 2003
8	Guo X X . Characterizations of disjointness of g-frames and constructions of g-frames in Hilbert spaces. Complex Anal Oper Theory, 2014, 8, 1547- 1563 doi: 10.1007/s11785-014-0364-4
9	Casazza P G . The art of frame theory. Taiwanese J Math, 2000, 4, 129- 201 doi: 10.11650/twjm/1500407227
10	Zhang W . Dual and aroximately dual Hilbert-Schmidt frames in Hilbert spaces. Results in Mathematics, 2018, 73 (4): 1- 20
11	Casazza P G , Kutyniok G , Li S . Fusion frames and distributed processing. Appl Comput Harmon Anal, 2008, 25, 114- 132 doi: 10.1016/j.acha.2007.10.001
12	Fornasier M . Quasi-orthogonal decompositions of structured frames. J Math Anal Appl, 2004, 289, 180- 199 doi: 10.1016/j.jmaa.2003.09.041
13	Li S , Ogawa H . Pseudo-frames for subspaces with applications. J Fourier Anal Appl, 2004, 10, 409- 431 doi: 10.1007/s00041-004-3039-0
14	Sun W C . G-frames and g-Riesz bases. J Math Anal Appl, 2006, 322, 437- 452 doi: 10.1016/j.jmaa.2005.09.039
15	Han D , Larson D . Frames, bases and group representations. Mem Amer Math Soc, 2000, 697, 94- 147
16	Sun W C . Stability of g-frames. J Math Anal Appl, 2007, 326, 858- 868 doi: 10.1016/j.jmaa.2006.03.043
17	Najati A , Faroughi M H , Rahimi A . G-frames and stability of g-frames in Hilbert spaces. Methods Funct Anal Topology, 2008, 14, 271- 286