K-框架和紧K-框架的算子扰动的稳定性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (1): 29-38.

K-框架和紧K-框架的算子扰动的稳定性

杜丹丹(),朱玉灿*()

福州大学数学与计算机科学学院福州 350116

收稿日期:2019-10-08 出版日期:2021-02-26 发布日期:2021-01-29
通讯作者: 朱玉灿 E-mail:1277702125@qq.com;zhuyucan@fzu.edu.cn
作者简介:杜丹丹, E-mail: 1277702125@qq.com
基金资助:
福建省自然科学基金(2016J01014);福建省自然科学基金(2020J01496)

Stabilities of K- Frames and Tight K- Frames Under the Operator Perturbation

Dandan Du(),Yucan Zhu*()

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116

Received:2019-10-08 Online:2021-02-26 Published:2021-01-29
Contact: Yucan Zhu E-mail:1277702125@qq.com;zhuyucan@fzu.edu.cn
Supported by:
Supported by the Natural Science Foundation of Fujian Province(2016J01014);Supported by the Natural Science Foundation of Fujian Province(2020J01496)

摘要/Abstract

摘要：

该文在Hilbert空间中讨论K-框架和紧K-框架在算子扰动中的稳定性.首先给出K-框架经过有界线性算子 $T $扰动后为K-框架的充要条件，其次讨论了用两个Bessel序列或者两个K-框架构造新的K-框架的方法，最后给出用两个Bessel序列构造紧K-框架的充要条件.这些结果推广和改进了由Christensen和Casazza等得到的著名结果.

关键词: K-框架, 紧K-框架, Bessel序列, 稳定性

Abstract:

In this paper, we discuss the stabilities of K-frames and tight K-frames under the operator perturbation. Firstly, we provide an equivalent characterization of the operator perturbation for a K-frame by using a bounded linear operator $T $ from ${{\cal H}_1} $ to ${{\cal H}_2} $. We also give a simple way to construct new K-frames from two existing Bessel sequences. Meanwhile, we make a discussion on the construction for K-frames from given ones. In the end, we obtain a necessary and sufficient condition to generate tight K-frames from two old Bessel sequences. Our results generalize and improve the remarkable results which had been obtained by Casazza and Christensen.

Key words: K- frame, Tight K- frame, Bessel sequence, Stability

中图分类号:

O171.1

杜丹丹,朱玉灿. K-框架和紧K-框架的算子扰动的稳定性[J]. 数学物理学报, 2021, 41(1): 29-38.

Dandan Du,Yucan Zhu. Stabilities of K- Frames and Tight K- Frames Under the Operator Perturbation[J]. Acta mathematica scientia,Series A, 2021, 41(1): 29-38.

参考文献

1	Poon C . A consistent and stable approach to generalized sampling. J Fourier Anal Appl, 2014, 20 (5): 985- 1019
2	Tsiligianni E V , Kondi L P , Katsaggelos A K . Construction of incoherent unit norm tight frames with application to compressed sensing. IEEE Trans Inform Theory, 2014, 60 (4): 2319- 2330
3	Christensen O . An Introduction to Frames and Riesz Bases Boston: Birkh?user, 2016
4	Casazza P G . The art of frame theory. Taiwanese J of Math, 2000, 4 (2): 129- 201
5	G?vru?a L . Frames for operators. Appl Comput Harmon Anal, 2012, 32 (1): 139- 144
6	朱玉灿, 舒志彪, 肖祥春. 复Hilbert空间中的K-框架和K-Riesz基. 中国科学:数学, 2018, 48 (5): 609- 622
6	Zhu Y C , Shu Z B , Xiao X C . K-frames and K-Riesz bases in complex Hilbert spaces. Sci Sin Math, 2018, 48 (5): 609- 622
7	Ramu G , Johnson P S . Frame operators of K-frames. SeMA Journal, 2016, 73 (2): 171- 181
8	Guo X X . Canonical dual K-Bessel sequences and dual K-Bessel generators for unitary systems of Hilbert spaces. J Math Anal Appl, 2016, 444 (1): 598- 609
9	Xiao X C , Zhu Y C , G?vru?a L . Some properties of K-frames in Hilbert spaces. Results Math, 2013, 63 (3/4): 1243- 1255
10	Jia M, Zhu Y C. Some results about the operator perturbation of a K-frame. Results Math, 2018, 73(4): Article number: 138
11	丁明玲, 肖祥春, 曾晓明. Hilbert空间中的紧K-框架. 数学学报, 2013, 56 (1): 105- 112
11	Ding M L , Xiao X C , Zeng X M . Tight K-frames in Hilbert spaces. Acta Math Sin, 2013, 56 (1): 105- 112
12	Najati A , Abdollahpour M R , Osgooei E , Saem M M . More sums of Hilbert space frames. Bull Kore Math Soc, 2013, 50 (6): 1841- 1846
13	Obeidat S , Samarah S , Casazza P G , Tremain J C . Sums of Hilbert space frames. J Math Anal Appl, 2009, 351 (2): 579- 585
14	Guo X X . Constructions of frames by disjoint frames. Numer Funct Anal Optim, 2014, 35 (5): 576- 587
15	郭训香. 框架的强分离性与紧框架的构造. 数学学报, 2015, 58 (4): 585- 592
15	Guo X X . Strong disjointness of frames and constructions of tight frames. Acta Math Sin, 2015, 58 (4): 585- 592
16	Douglas R G . On majorization, factorization, and range inclusion of operators on Hilbert space. Proc Amer Math Soc, 1966, 17 (2): 413- 415

[1]	王春, 许天周. 空间 $A^2$ 和 $\mathcal{D}^2$ 上具有 Hyers-Ulam 稳定性的系数乘子[J]. 数学物理学报, 2025, 45(5): 1381-1391.
[2]	邓钧元, 张峻皓. 半空间中满足 Cattaneo 定律的双曲系统的稳定性[J]. 数学物理学报, 2025, 45(5): 1444-1462.
[3]	吕东霆. 两种群空间非均匀反应扩散竞争模型的全局渐近稳定性[J]. 数学物理学报, 2025, 45(4): 1171-1183.
[4]	梁青. 两类带 Markov 切换的随机比例泛函微分方程全局解的存在唯一性和稳定性[J]. 数学物理学报, 2025, 45(4): 1184-1205.
[5]	刘佳, 包雄雄. 非局部时滞扩散方程棱锥形波前解的渐近稳定性[J]. 数学物理学报, 2025, 45(1): 44-53.
[6]	肖江龙, 宋永利, 夏永辉. 一个扩散模型中恐惧效应与集群行为协同诱导的时空动力学研究[J]. 数学物理学报, 2024, 44(6): 1577-1594.
[7]	张咏雪, 贾文生. 有限理性下广义多目标多主多从博弈受控系统解的稳定性[J]. 数学物理学报, 2024, 44(6): 1689-1702.
[8]	樊天娇, 冯立超, 杨艳梅. 不等式的推广以及在加性时变时滞系统中的应用[J]. 数学物理学报, 2024, 44(5): 1335-1351.
[9]	杨红, 张晓光. 单纯复形上的随机 SIS 传染病模型[J]. 数学物理学报, 2024, 44(5): 1392-1399.
[10]	郭燕, 徐小川. 基于混合谱数据的不连续Sturm-Liouville逆谱问题局部可解性和稳定性[J]. 数学物理学报, 2024, 44(4): 859-870.
[11]	潘丽君, 吕顺, 翁莎莎. 耦合Aw-Rascle-Zhang模型的Riemann解及其稳定性[J]. 数学物理学报, 2024, 44(4): 885-895.
[12]	章春国, 孙宝楠, 付煜之, 于欣. 具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定[J]. 数学物理学报, 2024, 44(4): 946-959.
[13]	高彩霞, 赵东霞. 带干扰项的 ARZ 交通流模型的时滞控制与 ISS 稳定性[J]. 数学物理学报, 2024, 44(4): 960-977.
[14]	徐瑞, 周凯娟, 白宁. 一类基于游离病毒感染和细胞-细胞传播的宿主体内 HIV-1 感染动力学模型[J]. 数学物理学报, 2024, 44(3): 771-782.
[15]	邱仰聪, 王其如. 一阶非线性时标动态方程的 Hyers-Ulam-Rassias 稳定性[J]. 数学物理学报, 2024, 44(2): 465-475.