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On weaving Hilbert space frames and Riesz bases

Part of: Topological linear spaces and related structures Nontrigonometric harmonic analysis General theory of linear operators

Published online by Cambridge University Press:  22 July 2025

Animesh Bhandari Open the ORCID record for Animesh Bhandari [Opens in a new window]
Animesh Bhandari*
Affiliation:
Department of Mathematics, SRM University, AP - Andhra Pradesh , Neeru Konda, Mangalagiri, Amaravati, Andhra Pradesh - 522240, India
*
e-mail: bhandari.animesh@gmail.com animesh.b@srmap.edu.in
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Abstract

Two frames $\{f_n\}_{n =1}^{\infty }$ and $\{g_n\}_{n =1}^{\infty }$ in a separable Hilbert space ${\mathcal H}$ are said to be weaving frames, if for every $\sigma \subset \mathbb N$, $\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n\in \sigma ^c}$ is a frame for ${\mathcal H}$. Weaving frames are proved to be very useful in many areas, such as, distributed processing, wireless sensor networks, packet encoding, and many more. Inspired by the work of Bemrose et al. [2], this paper delves into the properties and characterizations of weaving frames and weaving Riesz bases.

Keywords

FrameBesselian frameweaving frames

MSC classification

Primary: 42C15: General harmonic expansions, frames
Secondary: 46A32: Spaces of linear operators; topological tensor products; approximation properties 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 10
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Arefijamaal, A. and Zekaee, E., Image processing by alternate dual Gabor frames . Bull. Iran. Math. Soc. 42(2016), 1305–1314.Google Scholar
Bemrose, T., Casazza, P. G., Gröchenig, K., Lammers, M. C., and Lynch, R. G., Weaving frames . Oper. Matr. 10(2016), 1093–1116.10.7153/oam-10-61CrossRefGoogle Scholar
Bhandari, A. and Mukherjee, S., Atomic subspaces for operators . Indian J. Pure Appl. Math. 51(2020), 1039–1052.10.1007/s13226-020-0448-yCrossRefGoogle Scholar
Casazza, P. G., Freeman, D., and Lynch, R. G., Weaving Schauder frames . J. Approx. Theory 211(2016), 42–60.CrossRefGoogle Scholar
Casazza, P. G. and Kutyniok, G., Finite frames: Theory and applications, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, 2012.Google Scholar
Casazza, P. G. and Kutyniok, G., Frames of subspaces . Contemp. Math. 345(2014), 87–114.CrossRefGoogle Scholar
Casazza, P. G., Kutyniok, G., and Li, S., Fusion frames and distributed processing . Appl. Comput. Harmon. Anal. 25(2008), 114–132.CrossRefGoogle Scholar
Casazza, P. G., Kutyniok, G., Li, S., and Rozell, C. J., Modeling sensor networks with fusion frames . In: Proc. SPIE, vol. 6701, Wavelets XII, 67011M (20 September 2007); San Diego, California, 2007. https://doi.org/10.1117/12.730719.CrossRefGoogle Scholar
Christensen, O., An introduction to frames and Riesz bases, Birkhäuser, Boston, MA, 2003.10.1007/978-0-8176-8224-8CrossRefGoogle Scholar
Daubechies, I., Grossmann, A., and Meyer, Y., Painless non orthogonal expansions . J. Math. Phys. 27(1986), 1271–1283.10.1063/1.527388CrossRefGoogle Scholar
Deepshikha, and Vashisht, L. K., Weaving $K$ -frames in Hilbert spaces. Results Math. 73(2018), 81.10.1007/s00025-018-0843-4CrossRefGoogle Scholar
Duffin, R. and Schaeffer, A. C., A class of non harmonic Fourier series . Trans. Amer. Math. Soc. 72(1952), 341–366.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J., Linear operators, I, Interscience, New York, NY, 1963.Google Scholar
Ferreira, P., Mathematics for multimedia signal processing II: discrete finite frames and signal reconstruction, Signal Processing for Multimedia, Byrnes, J. S. (ed.), 1999, 35–54.Google Scholar
Gǎvruţa, L., Frames for operators . Appl. Comput. Harmon. Anal. 32(2012), 139–144.10.1016/j.acha.2011.07.006CrossRefGoogle Scholar
Gröchenig, K., Foundations of time-frequency analysis, Birkhäuser, New York, NY, 2001.10.1007/978-1-4612-0003-1CrossRefGoogle Scholar
Gumber, A. and Shukla, N. K., Uncertainty principle corresponding to an orthonormal wavelet system . Appl. Anal. 97(2018), 486–498.CrossRefGoogle Scholar
Han, D. and Larson, D. R., Frames, bases and group representations, Memoirs of the American Mathematical Society, 147(2000), 94 pp.Google Scholar
Holub, J. R., Pre-frame operators, Besselian frames, and near-Riesz BASES in Hilbert spaces . Proc. Amer. Math. Soc. 122(1994), 779–785.10.1090/S0002-9939-1994-1204376-4CrossRefGoogle Scholar
Retherford, J. and Holub, J., The stability of bases in Banach and Hilbert spaces . J. Reine Angew. Math. 246(1994), 136–146.Google Scholar
Schatten, J., Norm ideals of completely continuous operators, Springer-Verlag, Berlin, 1970.10.1007/978-3-662-35155-0CrossRefGoogle Scholar
Singer, I., Bases in Banach spaces, I, Springer-Verlag, Berlin, 1970.10.1007/978-3-642-51633-7CrossRefGoogle Scholar
Sun, W., $G$ -frames and $G$ -Riesz bases . J. Math. Anal. Appl. 322(2006), 437–452.10.1016/j.jmaa.2005.09.039CrossRefGoogle Scholar
Verma, G., Vashisht, L. K., and Singh, M., On excess of retro Banach frames . J. Contemp. Math. Anal. (Armenian Acad. Sci.) 54(2019), 143–146.Google Scholar
Xu, Y. and Leng, J., New inequalities for weaving frames in Hilbert spaces . J. Inequal. Appl. 188(2021), 15 pp.Google Scholar