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.

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

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