Abstract
Frame theory is an exciting, dynamic and fast paced subject with applications in numerous fields of mathematics and engineering. In this paper we study Continuous Frame and introduce Continuous Frame with \(C^{*}\)-valued bounds. Also, we establich some properties.
Similar content being viewed by others
Data availablity
Not applicable.
References
Ali, S.T., J.P. Antoine, and J.P. Gazeau. 1993. Continuous frames in hilbert space. Ann. Phys. (N. Y). 222: 1–37. https://doi.org/10.1006/aphy.1993.1016.
Alijani, A., and M.A. Dehghan. 2011. \(\ast \)-frames in hilbert \(C^{\ast }\)-modules. UPB Sci. Bull. Ser. A Appl. Math. Phys. 73: 89–106.
Arambašić, L. 2007. On frames for countably generated Hilbert \(C^{\ast }\)-modules. Proc. Am. Math. Soc. 135: 469–478. https://doi.org/10.1090/s0002-9939-06-08498-x.
Askari-Hemmat, A., M. Dehghan, and M. Radjabalipour. 2000. Generalized frames and their redundancy. Proc. Am. Math. Soc. 129: 1143–1147. https://doi.org/10.1090/s0002-9939-00-05689-6.
Christensen, O. 2016. An introduction to frames and riesz bases.
Christopher Lance, E., and L. Mathematical Society. 1995. Hilbert C\(\ast \)-Modules: a toolkit for operator algebraists - Page 35. Cambridge University Press.
Christopher Lance, E. 1995. Hilbert C\(\ast \)-Modules: a toolkit for operator algebraists.
Daubechies, I., A. Grossmann, and Y. Meyer. 1986. Painless nonorthogonal expansions. J. Math. Phys. 27: 1271–1283. https://doi.org/10.1063/1.527388.
Davidson, K.R. 1996. \(C^{\ast }\)-Algebras by Example. Amer Mathematical Society.
Duffin, R.J., and A.C. Schaeffer. 1952. A Class of Nonharmonic Fourier Series. Trans. Am. Math. Soc. 72: 341. https://doi.org/10.2307/1990760.
Gabardo, J.P., and D. Han. 2003. Frames associated with measurable spaces. Adv. Comput. Math. 18: 127–147. https://doi.org/10.1023/A:1021312429186.
Gabor, D. 1946. Theory of communications. J. Inst. Electr. Eng. 93: 429–457.
Jing, W. 2006. Frames in Hilbert C\(\ast \)-modules, Ph. D. Thesis, University of Central Florida.
Kaiser, G. 2011. A Friendly Guide to Wavelets. Boston: Birkhäuser Boston.
Kaplansky, I. 1953. Modules Over Operator Algebras. Am. J. Math. 75: 839. https://doi.org/10.2307/2372552.
Moosavi, Z.A., and A. Nazari. 2019. Controlled \(\ast \)-G-Frames and \(\ast \)-G-Multipliers in Hilbert Pro-C\(\ast \)-Modules. Int. J. Anal. Appl.https://doi.org/10.28924/2291-8639-17-2019-1.
Murphy, G.J. 1990. \(C^\ast \)-Algebras and Operator Theory. San Diego: California, Academic Press.
Paschke, W.L. 1973. Inner product modules over \(C^{\ast }\)-algebras. Trans. Am. Math. Soc. 182: 443–468. https://doi.org/10.1090/s0002-9947-1973-0355613-0.
Rahimi, A., A. Najati, and Y.N. Dehghan. 2006. Continuous frame in Hilbert spaces. Methods Funct. Anal. Topol. 12: 170–182.
Rossafi, M.A.B., H. Labrigui, and A. Touri. 2020. The duals of \(\ast \)-operator Frame for \(End_{\cal{A}}^{\ast }(\cal{H})\). Asia Math 4: 45–52.
Rossafi, M., and A. Akhlidj. 2018. Perturbation and Stability of Operator Frame for \(End_{\cal{A} }^{\ast }(\cal{H} )\). Math-Recherche Appl. 4: 45–52.
Rossafi, M., and S. Kabbaj. 2018. \(\ast \)-g-frames in tensor products of Hilbert \(C^{\ast }\) -modules. Ann. Univ. Paedagog. Cracoviensis. Stud. Math. 17: 17–25. https://doi.org/10.2478/aupcsm-2018-0002.
Rossafi, M., and S. Kabbaj. 2020. \(\ast \)-K-operator frame for End\(\ast \)A(H). Asian-Euro. J. Math. 13: 2050060. https://doi.org/10.1142/S1793557120500606.
Rossafi, M., and S. Kabbaj. 2018. Frames and Operator Frames for \(B(H)\). Asia Math. 2: 19–23.
Rossafi, M., and S. Kabbaj. 2019. Operator Frame for \(End_{\cal{A} }^{\ast }(\cal{H} )\). J. Linear Topol. Algebr. 8: 85–95.
Wiener, N., and R. Paley. 1934. Fourier transforms in the complex domain. Providence, Rhode Island: American Mathematical Society.
Xu, Q., and L. Sheng. 2008. Positive semi-definite matrices of adjointable operators on Hilbert \(C^{\ast }\)-modules. Linear Algebra Appl. 428: 992–1000. https://doi.org/10.1016/j.laa.2007.08.035.
Yosida, K. 1978. Functional analysis. Berlin Heidelberg, Berlin, Heidelberg: Springer.
Acknowledgements
It is our great pleasure to thank the referee for his careful reading of the paper and for several helpful suggestions.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All authors contributed substantially to this paper, participated in drafting and checking the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Communicated by S Ponnusamy.
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
Rossafi, M., Ghiati, M., Mouniane, M. et al. Continuous frame in Hilbert \(C^{*}\)-modules. J Anal 31, 2531–2561 (2023). https://doi.org/10.1007/s41478-023-00581-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-023-00581-8