Characterization of Dual Scalable Frames

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.