Abstract
Weaving Hilbert space frames have been introduced recently by Bemrose et al. to deal with some problems in distributed signal processing. In this paper, we survey this topic from the viewpoint of the duality principle. In this regard, not only we obtain new properties in weaving frame theory related to dual frames but also we bring up new approaches for manufacturing pairs of woven frames. Specifically, we give some sufficient conditions under which a frame with its canonical dual, alternate duals or approximate duals constitute some concrete pairs of woven frames. Moreover, we provide some approaches for constructing weaving frames by using small perturbations and present a condition where different operators preserve the weaving property. As a consequence, the canonical duals of two woven frames are woven.
Similar content being viewed by others
References
Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)
Arabyani-Neyshaburi, F., Arefijamaal, A.: Some constructions of \(K\)-frames and their duals. Rocky Mt. J. Math. 47(6), 1749–1764 (2017)
Arefijamaal, A., Zekaee, E.: Signal processing by alternate dual gabor frames. Appl. Comput. Harmon. Anal. 35, 535–540 (2013)
Arefijamaal, A., Arabyani-Neyshaburi, F.: Some properties of dual and approximate dual of fusion frames. Turk. J. Math. 41, 1191–1203 (2017)
Balan, R., Casazza, P.G., Heil, C., Landau, Z.: Deficits and excesses of frames. Adv. Comput. Math. 18, 93–116 (2003). (Special issue on frames)
Bemrose, T., Casazza, P.G., Grochenig, K., Lammers, M.C., Lynch, R.G.: Weaving Hilbert space frames. Oper. Matrices 10(4), 1093–1116 (2016)
Benedetto, J., Powell, A., Yilmaz, O.: Sigm-Delta quantization and finite frames. IEEE Trans. Inf. Theory 52, 1990–2005 (2006)
Bodmann, B.G., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258(2), 397–420 (2010)
Bodmann, B.G., Paulsen, V.I.: Frames, graphs and erasures. Linear Algebra Appl. 404, 118–146 (2005)
Bolcskel, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46, 3256–3268 (1998)
Cahill, J., Casazza, P.G., Kutyniok, G.: Operators and frames. J. Oper. Theory 70(1), 145–164 (2013)
Candes, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities. Commun. Pure Appl. Anal. 56, 216–266 (2004)
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 the SampTA, pp. 110–114 (2015)
Christensen, O.: Frames and Bases: An Introductory Course. Birkhäuser, Boston (2008)
Christensen, O., Laugesen, R.S.: Approximately dual frames in Hilbert spaces and applications to Gabor frames. Sampl. Theory Signal Image Process. 9, 77–89 (2010)
Ghaani Farashahi, A.: Square-integrability of multivariate metaplectic wave-packet representations. J. Phys. A Math. Theor. 50(115202), 1–22 (2017)
Ghaani Farashahi, A.: Square-integrability of metaplectic wave-packet representations on \(L^{2}(R)\). J. Math. Anal. Appl. 449(1), 769–79 (2017)
Ghaani Farashahi, A.: Multivariate wave-packet transforms. J. Anal. Appl. 36(4), 481–500 (2017)
Găvruţa, L.: Frames for operators. Appl. Comput. Harmon. Anal. 32, 139–144 (2012)
Javanshiri, H.: Some properties of approximate dual frames in Hilbert spaces. Results Math. 70, 475–485 (2016)
Kaftal, V., Larson, D.R., Zhang, Sh: Operator-valued frames. Trans. Am. Math. Soc. 361, 6349–6385 (2009)
Sun, W.: G-frames and g-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
Vashisht, L.K., Deepshikha: On weaving frames. Houston J. Math. (to appear)
Vashisht, L.K., Deepshikha, Garg, S., Daus, P.K.: On genralized weaving frames of Hilbert spaces. Rocky Mt. J. Math. (to appear)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Sorina Barza.
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
Arabyani-Neyshaburi, F., Arefijamaal, A.A. Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces. Bull. Malays. Math. Sci. Soc. 44, 147–161 (2021). https://doi.org/10.1007/s40840-020-00940-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-00940-9