Abstract
Weaving frames have potential applications in wireless sensor networks that require distributed processing of signals under different frames. In this paper, we study some new properties of weaving generalized frames (or g-frames) and weaving generalized orthonormal bases (or g-orthonormal bases). It is shown that a g-frame and its dual g-frame are woven. The inter-relation of optimal g-frame bounds and optimal universal g-frame bounds is studied. Further, we present a characterization of weaving g-frames. Illustrations are given to show the difference in properties of weaving generalized Riesz bases and weaving Riesz bases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bemrose, T., Casazza, P.G., Gröchenig, K., Lammers, M.C., Lynch, R.G.: Weaving frames. Oper. Matrices 10(4), 1093–1116 (2016)
Casazza, P.G., Freeman, D., Lynch, R.G.: Weaving Schauder frames. J. Approx. Theory 211, 42–60 (2016)
Casazza, P.G., Kutyniok, G.: Finite Frames: Theory and Applications. Birkhäuser, Basel (2012)
Casazza, P.G., Kutyniok, G.: Frames of subspaces. Contemp. Math. 345, 87–114 (2004)
Casazza, P.G., Lynch, R.G.: Weaving properties of Hilbert space frames. In: Proceedings of SampTA, pp. 110–114 (2015)
Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Basel (2016)
Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)
Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
Deepshikha, Vashisht, L.K.: On weaving frames. Houston J. Math. 44(3), 887–915 (2018)
Deepshikha, Vashisht, L.K.: Vector-valued (super) weaving frames. J. Geom. Phys. 134, 48–57 (2018)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Feichtinger, H., Strohmer, T.: Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Basel (1998)
Fornasier, M.: Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl. 289, 80–199 (2004)
Han, D., Larson, D.R.: Frames, bases and group representations. Mem. Am. Math. Soc. 147(697), 1–94 (2000)
Khosravi, A., Banyarani, J.S.: Weaving \(g\)-frames and weaving fusion frames. Bull. Malays. Math. Sci. Soc. 42(6), 3111–3129 (2019)
Khosravi, A., Musazadeh, K.: Fusion frames and \(g\)-frames. J. Math. Anal. Appl. 342(2), 1068–1083 (2008)
Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10, 409–431 (2004)
Sun, W.: \(g\)-frames and \(g\)-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
Vashisht, L.K., Deepshikha: Weaving properties of generalized continuous frames generated by an iterated function system. J. Geom. Phys. 110, 282–295 (2016)
Vashisht, L.K., Garg, S., Deepshikha, Das, P.K.: On generalized weaving frames in Hilbert spaces. Rocky Mountain J. Math. 48(2), 661–685 (2018)
Acknowledgements
The authors thank the anonymous referee for a careful and thorough reading of the paper and for valuable comments. The authors are grateful to Dr. Lalit Kumar Vashisht, University of Delhi, for his valuable comments and suggestions, which improved the presentation of the paper. Aniruddha Samanta thanks University Grants Commission (UGC) for the financial support in the form of the Senior Research Fellowship (Ref. No: 19/06/2016(i)EU-V; Roll No. 423206).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yoshihiro Sawano.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Deepshikha, Samanta, A. On Weaving Generalized Frames and Generalized Riesz Bases. Bull. Malays. Math. Sci. Soc. 45, 361–378 (2022). https://doi.org/10.1007/s40840-021-01193-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-021-01193-w