Abstract
In this paper, using some real numbers as parameters, we introduce some new versions of resolutions of the identity, atomic systems and frame-like systems for subspaces of a Hilbert space. The new versions cover many of the notions related to atomic systems and resolutions of the identity, also, the existence of the parameters provides more flexible tools for the reconstruction of signals. It is shown that there are close relationships between the new notions and some generalizations of frames and fusion frames. Moreover, some properties and applications of the new concepts are obtained, especially their stability under perturbations and direct sums is considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
The manuscript has no associated data
References
Abdollahi, A., Rahimi, E.: Generalized frames on super Hilbert spaces. Bull. Malays. Math. Sci. Soc. 35, 807–818 (2012)
Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)
Asgari, M.S., Khosravi, A.: Frames and bases of subspaces in Hilbert spaces. J. Math. Anal. Appl. 308, 541–553 (2005)
Balazs, P.: Basic definition and properties of Bessel multipliers. J. Math. Anal. Appl. 325, 571–585 (2007)
Balazs, P., Bayer, D., Rahimi, A.: Multipliers for continuous frames in Hilbert spaces. J. Phys. A. 45, 244023–244043 (2012)
Casazza, P., Kutyniok, G.: Frames of subspaces. Contemp. Math. Am. Math. Soc. 345, 87–113 (2004)
Christensen, O., Stoeva, D.T.: p-frames in separable Banach spaces. Adv. Comput. Math. 18, 117–126 (2003)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Faroughi, M.H., Ahmadi, R.: Some properties of C-fusion frames. Turk. J. Math. 34, 393–415 (2010)
Feichtinger, H.G., Werther, T.: Atomic system for subspaces. In: Zayed, L. (ed.) Proceedings SampTA 2001, Orlando, pp. 163–165 (2001)
Gabardo, J.P., Han, D.: Frame associated with measurable spaces. Adv. Comput. Math. 18, 127–147 (2003)
Javanshiri, H., Fattahi, A.M.: Continuous atomic systems for subspaces. Mediterr. J. Math. 13, 1871–1884 (2016)
Kaiser, G.: A Friendly Guide to Wavelets. Birkhauser, Boston (1994)
Khosravi, A., Asgari, M.S.: Frames of subspaces and approximation of the inverse frame operator. Houst. J. Math. 33, 907–920 (2007)
Khosravi, A., Mirzaee Azandaryani, M.: Fusion frames and g-frames in tensor product and direct sum of Hilbert spaces. Appl. Anal. Discrete Math. 6, 287–303 (2012)
Khosravi, A., Mirzaee Azandaryani, M.: G-frames and direct sums. Bull. Malays. Math. Sci. Soc. 36, 313–323 (2013)
Mirzaee Azandaryani, M., Javadi, Z.: Pseudo-duals of continuous frames in Hilbert spaces. J. Pseudo-Differ. Oper. Appl. 13, 1–16 (2022)
Author information
Authors and Affiliations
Contributions
Conceptualization, supervision, validation and editing by MMA, formal analysis, investigation and writing—original draft by ZAMA.
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mirzaee Azandaryani, M., Aghamir Mohammad Ali, Z. Atomic and Frame-Like Systems for Subspaces of a Hilbert Space. Mediterr. J. Math. 20, 185 (2023). https://doi.org/10.1007/s00009-023-02395-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-023-02395-1