Frame properties of systems arising via iterated actions of operators

By Ole W. Christensen, Marzieh Hasannasab. Applied and Computational Harmonic Analysis, 2019

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Abstract 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 { ( T j ) n φ j } n = 0 ∞ , j = 1 , … , J for some bounded operators T j and elements φ j 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 φ 1 , … , φ J by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system { T n φ } n = 0 ∞ , 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.

Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/e22548660b9856adfa5a625e4e10b310d1283607