Abstract
Scalable frames in separable Hilbert spaces have been recently introduced by Kutyniok et al. to modify a general frame and to generate a Parseval frame by rescaling frame vectors. The main framework proposed in this paper is based on the redundancy of frame elements and is used as input for classification. This method leads to a complete characterization of scalable frames in \(\mathbb {R}^{2}\) and \(\mathbb {R}^{3}\). In addition, we introduce all possible choices for the scale coefficients of a given scalable frame. Finally, we discuss the scalability of duals frames. We divide the set of all scalable dual frames of a given frame into two disjoint subsets, containing and not containing an orthogonal basis. In particular, we prove that both of them are non-empty.
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Acknowledgements
The first and the second authors were supported in part by the Hakim Sabzevari university (HSU) under Grant 943958. The authors also thank Fahimeh Arabyani Neyshaburi for fruitful discussions.
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This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Andreas Seeger, Franz Luef and Serap Oztop.
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Heydarpour, B., Arefijamaal, A.A. & Ghaani Farashahi, A. Characterization of Dual Scalable Frames. Complex Anal. Oper. Theory 18, 65 (2024). https://doi.org/10.1007/s11785-024-01516-2
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DOI: https://doi.org/10.1007/s11785-024-01516-2