American Institute of Mathematical Sciences

Advanced
May  2020, 3(2): 101-116. doi: 10.3934/mfc.2020008

Weaving K-fusion frames in hilbert spaces

Hanbing Liu 1, , Yongdong Huang 1,2,, and  Chongjun Li 3,

1. 

School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China
2. 

School of Science, Dalian Minzu University, Dalian, 116600, China
3. 

School of Mathematics Science, Dalian University of Technology, Dalian, 116024, China

* Corresponding author: Yongdong Huang

Received  December 2019 Revised  April 2020 Published  May 2020

$ K $-fusion frames are generalizations of fusion frames in frame theory. In this paper, based on the weaving frames and $ K $-fusion frames, we propose the notion of weaving $ K $-fusion frames and conduct relevant research. First, we give some characterizations of weaving $ K $-fusion frames. Then, by means of operator theory and frame theory, we present several novel construction approaches of weaving $ K $-fusion frames. Finally, we discuss transitivity of weaving $ K $-fusion frames.

Keywords: Fusion frame, Bessel sequence, weaving frame, atomic system, $ K $-fusion frame.
Mathematics Subject Classification: Primary: 42C15; Secondary: 42C30, 42C40.
Citation: Hanbing Liu, Yongdong Huang, Chongjun Li. Weaving K-fusion frames in hilbert spaces. Mathematical Foundations of Computing, 2020, 3 (2) : 101-116. doi: 10.3934/mfc.2020008
References:
[1]

T. Bemrose, P. G. Casazza, K. Gröchenig, M. C. Lammers and R. G. Lynch, Weaving frames, Oper. Matrices, 10 (2016), 1093-1116.  doi: 10.7153/oam-10-61.  Google Scholar

[2]

P. G. Casazza, The art of frame theory, Taiwanese J. Math., 4 (2000), 129-202.  doi: 10.11650/twjm/1500407227.  Google Scholar

[3]

P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 25 (2008), 114-132.  doi: 10.1016/j.acha.2007.10.001.  Google Scholar

[4]

P. G. Casazza and G. Kutyniok, Frames of subspaces, Wavelets, Frames and Operator Theory, 345 (2004), 87-113.  doi: 10.1090/conm/345/06242.  Google Scholar

[5]

O. Christensen, An Introduction to Frames and Riesz Bases, 2$^nd$ edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-25613-9.  Google Scholar

[6]

I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), 1271-1283.  doi: 10.1063/1.527388.  Google Scholar

[7]

S. G. Deepshikha, L. K. Vashisht and G. Verma, On weaving fusion frames for Hilbert spaces, 2017 International Conference on Sampling Theory and Applications (SampTA), Tallin, (2017), 381–385. doi: 10.1109/SAMPTA.2017.8024363.  Google Scholar

[8]

V. Deepshikha and L. K. Vashisht, Weaving $K$-frames in Hilbert spaces, Results. Math., 73 (2018), 20 pp. doi: 10.1007/s00025-018-0843-4.  Google Scholar

[9]

R. J. Duffin and A. C. Schaeffer, A class of nonharmonic fourier series, Trans. Amer. Math. Soc., 72 (1952), 341-366.  doi: 10.1090/S0002-9947-1952-0047179-6.  Google Scholar

[10]

P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math., 7 (1971), 254-281.  doi: 10.1016/S0001-8708(71)80006-3.  Google Scholar

[11]

D. Hua and Y. Huang, Controlled $K$-g-frames in Hilbert spaces, Results Math., 72 (2017), 1227-1238.  doi: 10.1007/s00025-016-0613-0.  Google Scholar

[12]

Y. Huang and S. Shi, New Constructions of $K$-g-frames, Results Math., 73 (2018), 13 pp. doi: 10.1007/s00025-018-0924-4.  Google Scholar

[13]

A. Khosravi and B. Khosravi, Fusion frames and $g$-frames in Hilbert $C^{\ast}$-modules, Int. J. Wavelets Multiresolut. Inf. Process., 6 (2008), 433-446.  doi: 10.1142/S0219691308002458.  Google Scholar

[14]

A. Khosravi and K. Musazadeh, Fusion frames and $g$-frames, J. Math. Anal. Appl., 342 (2008), 1068-1083.  doi: 10.1016/j.jmaa.2008.01.002.  Google Scholar

[15]

D. Li and J. Leng, Fusion frames for operators and atomic systems, preprint, arXiv: 1801.02785. Google Scholar

[16]

F. A. Neyshaburi and A. A. Arefijamaal, Characterization and construction of $K$-fusion frames and their duals in hilbert spaces, Results Math., 73 (2018), 26 pp. doi: 10.1007/s00025-018-0781-1.  Google Scholar

[17]

F. A. Neyshaburi and A. A. Arefijamaal, Weaving Hilbert space fusion frames, preprint, arXiv: 1802.03352. Google Scholar

[18]

A. Rahimi, Z. Samadzadeh and B. Daraby, Woven fusion frames in Hilbert spaces, preprint, arXiv: 1808.03765. Google Scholar

[19]

S. Shi and Y. Huang, $K$-g-frames and their dual, Int. J. Wavelets Multiresolut. Inf. Process., 17 (2019), 11 pp. doi: 10.1142/S0219691319500152.  Google Scholar

[20]

W. Sun, $G$-frames and $g$-Riesz bases, J. Math. Anal. Appl., 322 (2006), 437-452.  doi: 10.1016/j.jmaa.2005.09.039.  Google Scholar

show all references

References:
[1]

T. Bemrose, P. G. Casazza, K. Gröchenig, M. C. Lammers and R. G. Lynch, Weaving frames, Oper. Matrices, 10 (2016), 1093-1116.  doi: 10.7153/oam-10-61.  Google Scholar

[2]

P. G. Casazza, The art of frame theory, Taiwanese J. Math., 4 (2000), 129-202.  doi: 10.11650/twjm/1500407227.  Google Scholar

[3]

P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 25 (2008), 114-132.  doi: 10.1016/j.acha.2007.10.001.  Google Scholar

[4]

P. G. Casazza and G. Kutyniok, Frames of subspaces, Wavelets, Frames and Operator Theory, 345 (2004), 87-113.  doi: 10.1090/conm/345/06242.  Google Scholar

[5]

O. Christensen, An Introduction to Frames and Riesz Bases, 2$^nd$ edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-25613-9.  Google Scholar

[6]

I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), 1271-1283.  doi: 10.1063/1.527388.  Google Scholar

[7]

S. G. Deepshikha, L. K. Vashisht and G. Verma, On weaving fusion frames for Hilbert spaces, 2017 International Conference on Sampling Theory and Applications (SampTA), Tallin, (2017), 381–385. doi: 10.1109/SAMPTA.2017.8024363.  Google Scholar

[8]

V. Deepshikha and L. K. Vashisht, Weaving $K$-frames in Hilbert spaces, Results. Math., 73 (2018), 20 pp. doi: 10.1007/s00025-018-0843-4.  Google Scholar

[9]

R. J. Duffin and A. C. Schaeffer, A class of nonharmonic fourier series, Trans. Amer. Math. Soc., 72 (1952), 341-366.  doi: 10.1090/S0002-9947-1952-0047179-6.  Google Scholar

[10]

P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math., 7 (1971), 254-281.  doi: 10.1016/S0001-8708(71)80006-3.  Google Scholar

[11]

D. Hua and Y. Huang, Controlled $K$-g-frames in Hilbert spaces, Results Math., 72 (2017), 1227-1238.  doi: 10.1007/s00025-016-0613-0.  Google Scholar

[12]

Y. Huang and S. Shi, New Constructions of $K$-g-frames, Results Math., 73 (2018), 13 pp. doi: 10.1007/s00025-018-0924-4.  Google Scholar

[13]

A. Khosravi and B. Khosravi, Fusion frames and $g$-frames in Hilbert $C^{\ast}$-modules, Int. J. Wavelets Multiresolut. Inf. Process., 6 (2008), 433-446.  doi: 10.1142/S0219691308002458.  Google Scholar

[14]

A. Khosravi and K. Musazadeh, Fusion frames and $g$-frames, J. Math. Anal. Appl., 342 (2008), 1068-1083.  doi: 10.1016/j.jmaa.2008.01.002.  Google Scholar

[15]

D. Li and J. Leng, Fusion frames for operators and atomic systems, preprint, arXiv: 1801.02785. Google Scholar

[16]

F. A. Neyshaburi and A. A. Arefijamaal, Characterization and construction of $K$-fusion frames and their duals in hilbert spaces, Results Math., 73 (2018), 26 pp. doi: 10.1007/s00025-018-0781-1.  Google Scholar

[17]

F. A. Neyshaburi and A. A. Arefijamaal, Weaving Hilbert space fusion frames, preprint, arXiv: 1802.03352. Google Scholar

[18]

A. Rahimi, Z. Samadzadeh and B. Daraby, Woven fusion frames in Hilbert spaces, preprint, arXiv: 1808.03765. Google Scholar

[19]

S. Shi and Y. Huang, $K$-g-frames and their dual, Int. J. Wavelets Multiresolut. Inf. Process., 17 (2019), 11 pp. doi: 10.1142/S0219691319500152.  Google Scholar

[20]

W. Sun, $G$-frames and $g$-Riesz bases, J. Math. Anal. Appl., 322 (2006), 437-452.  doi: 10.1016/j.jmaa.2005.09.039.  Google Scholar

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