Abstract
In this paper, we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set E. Starting from a frame \((x_n)_{n=1}^\infty \) and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of \((x_n)_{n\in E^c}\) so that the perfect reconstruction can be obtained from the preserved frame coefficients. The work is motivated by methods using the canonical dual frame of \((x_n)_{n=1}^\infty \), which however do not extend automatically to the case when the canonical dual is replaced with another dual frame. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated. We also give several ways of computing a dual of the reduced frame, among which we are the most interested in the iterative procedure for computing this dual frame.
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Acknowledgements
The authors are grateful to the reviewers for their valuable comments and suggestions. The second author is grateful for the hospitality of the Department of Mathematics (Faculty of Science, University of Zagreb) during her visit.
Funding
Open access funding provided by University of Vienna. The authors are supported by the Scientific and Technological Cooperation project Austria–Croatia “Frames, Reconstruction, and Applications” (HR 03/2020). The first author was also supported by the Croatian Science Foundation under the project IP-2016-06-1046. The second author acknowledges support from the Austrian Science Fund (FWF) under grants P 35846-N (DOI 10.55776/P35846), P 32055-N31, and Y 551-N13, and from the Vienna Science and Technology Fund (WWTF) through the project VRG12-009.
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Arambašić, L., Stoeva, D. Dual frames compensating for erasures—a non-canonical case. Adv Comput Math 50, 9 (2024). https://doi.org/10.1007/s10444-023-10104-5
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DOI: https://doi.org/10.1007/s10444-023-10104-5