New aspects of weaving K-frames: the excess and duality

Year 2024, Volume: 53 Issue: 3, 652 - 666, 27.06.2024

Elahe Agheshteh Moghaddam Ali Akbar Arefijamaal

https://doi.org/10.15672/hujms.1008448

Abstract

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. Likewise weaving K -frames have been proved to be useful during signal reconstructions from the range of a bounded linear operator K. In this paper, we study the notion of weaving and its connection to the duality of K-frames and construct several pairs of woven K-frames. Also, we find a unique biorthogonal sequence for every K-Riesz basis and obtain a $K^*$-frame which is woven by its canonical dual. Moreover, we describe the excess for K-frames and prove that any two woven K-frames in a separable Hilbert space have the same excess. Finally, we introduce the necessary and sufficient condition under which a K-frame and its image under an invertible operator have the same excess.

Keywords

frames, dual frames, Riesz bases, woven frames, excess of frames, K-frames

References

• [1] E. Agheshteh Moghadam and A. Arefijamaal, On Excesses and Duality in Woven Frames, Bull. Malays. Math. Sci. Soc. 44 (5), 3361–3375, 2021.
• [2] A. Aldroubi, Portraits of Frames, Proc. Amer. Math. Soc. 123 (1), 1661–1668, 1995.
• [3] F. Arabyani Neyshaburi and A. Arefijamaal, Manufactoring Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces, Bull. Malays. Math. Sci. Soc. 44 (1), 147–161, 2020.
• [4] F. Arabyani Neyshaburi and A. Arefijamaal, Some construction of K-frames and their duals, Rocky Mt. J. Math. 47 (6), 1749–1764, 2017.
• [5] A. Arefijamaal and E. Zekaee, Image Processing by Alternate Dual Gabor Frames, Bull. Iranian Math. Soc. 42 (6), 1305–1314, 2016.
• [6] A. Arefijamaal and E. Zekaee, Signal Processing by Alternate Dual Gabor Frames, Appl. Comput. Harmon. Anal. 35, 535–540, 2013.
• [7] A. Bhandari, D. Borah and S. Mukherjee, Characterizations of Weaving K-frames, Proc. Japan Acad. Ser. A Math. Sci. 96 (5), 39–43, 2020.
• [8] D. Bakic, T. Beric, On excesses of frames, Glas. Mat. Ser. III 50 (2), 415–427, 2015.
• [9] P. Balazs and K. Gröchenig. A guide to localized frames and applications to galerkinlike representations of operators. In I. Pesenson, H. Mhaskar, A. Mayeli, Q. T. L. Gia, and D.-X. Zhou, editors, Novel methods in harmonic analysis with applications to numerical analysis and data processing, Applied and Numerical Harmonic Analysis series (ANHA). Birkhauser/Springer, 2017.
• [10] P. Balazs, M. Shamsabadi, A. Arefijamaal, and A. Rahimi, U-cross Gram matrices and their invertibility, J. Math. Anal. Appl. 476 (2), 367–390, 2019.
• [11] T. Bemrose, P. G. Casazza, K. Grochenig,M. C. Lammers and R. G Lynch, Weaving frames, Oper. Matrices 10 (4), 1093–1116, 2016.
• [12] H. Boleskel,F. Hlawatsch and H. G. Feichtinger , Frame Theoretic Analysis of Oversampled Filter Banks, IEEE Trans. Signal Process 46, 3256–3268, 1998.
• [13] P. G. Casazza and G. Kutyniok, Frames of Subspaces, Contempt. Math. 345, 87–113, 2004.
• [14] P. G. Casazza, The art of Frame Theory, Taiwanese J. Math. 4 (2), 129–202, 2000.
• [15] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston, 2008.
• [16] N. Cotfas and J. P. Gazeau. Finite tight frames and some applications, J. Phys. A: Math. Theor. 43 (19), 193001, 2010.
• [17] Deepshikha and Lalit K. Vashisht, Weaving K-Frames in Hilbert spaces, Results Math. 27, 73–81, 2018 .
• [18] R. Duffin and A. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Amer. Math. Soc. 72, 341–366, 1952.
• [19] Y.C. Eldar, O. Christensen, A characterization of oblique dual frame Pairs, EURASIP J. Adv. Sig. Proc. 1–11, 2006.
• [20] S. Garg and L.K. Vashisht, Weaving K-Fusion Frames in Hilbert Spaces, Ganita 67 (1), 41–52, 2017.
• [21] L. Gˇavruta, Frames for Operators, Appl. Comp. Harmon. Anal. Appl. 32, 139–144, 2012.
• [22] J.R. Holub, Pre-Frames Operators, Besselian Frames and near-Riesz Bases in Hilbert Spaces, Amer. Math. Soc. 122, 779–785, 1994.
• [23] Gh. Rahimlou, Weaving continuous K-frames in Hilbert spaces, Probl. Anal. Issues Anal. 11(29), 91-105, 2022.
• [24] G. Ramu and P. Sam Johnson, Frame Operator of K-frames, SeMA. J. 37 (2), 171-181, 2016.
• [25] M. Shamsabadi and A. Arefijamaal, Some results of K-Frames and their multipliers, Turk. J. Math. 44, 538–552, 2020.
• [26] M. Shamsabadi and A. A. Arefijamaal. Some results on U-cross Gram matrices by using K-frames, Afrika Matematika 31, 1349–1358, 2020.
• [27] Zhong-Qi Xiang, Some New Results of Weaving K-Frames in Hilbert Spaces, Numer. Funct. Anal. Optim. 42 (4), 1-23, 2021.
• [28] X. Xiao, Y. Zhu and L. Gˇavruta, Some Properties of K-Frames in Hilbert spaces, Results. Math. 63, 1243–1255, 2013.
• [29] X. Xiao, K. Yan, G. Zhao, et al. Tight K-frames and weaving of K-frames, J. Pseudo- Differ. Oper. Appl. 12, 1–14, 2021.