Korean J. Math. Vol. 29 No. 1 (2021) pp.41-55
DOI: https://doi.org/10.11568/kjm.2021.29.1.41

Woven $g$-frames in Hilbert $C^*$-modules

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Ekta Rajput
Nabin Kumar Sahu
Vishnu Narayan Mishra


Woven frames are motivated from distributed signal processing with potential applications in wireless sensor networks. g-frames provide more choices on analyzing functions from the frame expansion coefficients. The objective of this paper is to introduce woven $g$-frames in Hilbert $C^*$-modules, and to develop its fundamental properties. In this investigation, we establish sufficient conditions under which two g-frames possess the weaving properties. We also investigate the sufficient conditions under which a family of g-frames possess weaving properties.

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[1] L. Arambaic, On frames for countably generated Hilbert C∗-modules, Proc. Amer. Math. Soc. 135 (2007), 469–478. Google Scholar

[2] T. Bemrose, P. G. Casazza, K. Grochenig, M. C. Lammers and R. G. Lynch, Weaving frames, Oper. Matrices 10 (4) (2016), 1093–1116. Google Scholar

[3] H. Bolcskei, F. Hlawatsch and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process. 46 (12) (1998), 3256–3268. Google Scholar

[4] J. S. Byrnes, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, Signals Processing for Multimedia 174 (1999), 35–54. Google Scholar

[5] E. J. Candes and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities, Commun. Pure Appl. Math. 57 (2) (2004), 219–266. Google Scholar

[6] P. G. Casazza, G. Kutyniok and Sh. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal. 25 (2008), 114–132. Google Scholar

[7] P. G. Casazza and R. G. Lynch, Weaving properties of Hilbert space frames, Sampling Theory and Applications (SampTA) (2015), 110–114 IEEE. Google Scholar

[8] P. G. Casazza, D. Freeman and R. G. Lynch, Weaving Schauder frames, J. Approx. Theory 211 (2016), 42–60. Google Scholar

[9] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271–1283. Google Scholar

[10] R. J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. Google Scholar

[11] Y. C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier. Anal. Appl. 9 (1) (2003), 77–96. Google Scholar

[12] M. Frank and D. R. Larson, Frames in Hilbert C*-modules and C*-algebras, J. Operator Theory 48 (2002), 273–314. Google Scholar

[13] F. Ghobadzadeh, A. Najati, G. A. Anastassiou and C. Park, Woven frames in Hilbert C∗- modules, J. Comput. Anal. Appl. 25 (2018), 1220–1232. Google Scholar

[14] I. Kaplansky, Algebra of type I, Annals of Math. 56 (1952), 460–472. Google Scholar

[15] A. Khosravi and B. Khosravi, Fusion frames and g-frames in Hilbert C*-modules, Int. J. Wavelets Multiresol. Inf. Process. 6 (2008), 433–466. Google Scholar

[16] S. Li and H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl. 10 (4) (2004), 409–431. Google Scholar

[17] D. Li, J. Leng and T. Huang, On weaving g-frames for Hilbert spaces, Complex Anal. Oper. Theory 14 (2) (2017), 1–25. Google Scholar

[18] W. Paschke, Inner product modules over B∗-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. Google Scholar

[19] T. Strohmer and R. Jr. Heath, Grassmanian frames with applications to coding and communi- cations, Appl. Comput. Harmon. Anal. 14 (2003), 257–275. Google Scholar

[20] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (1) (2006), 437–452. Google Scholar