Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences View the MathML source for some bounded operators Tj and elements φj in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators φ1,…,φJ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system View the MathML source but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.
Christensen, O., & Hasannasabjaldehbakhani, M. (2018). Frame properties of systems arising via iterated actions of operators. Applied and Computational Harmonic Analysis, 46(3), 664-673. https://doi.org/10.1016/j.acha.2018.04.002