Frame properties of systems arising via iterated actions of operators

Ole Christensen*, Marzieh Hasannasabjaldehbakhani

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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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 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.
Original languageEnglish
JournalApplied and Computational Harmonic Analysis
Volume46
Issue number3
Pages (from-to)664-673
ISSN1063-5203
DOIs
Publication statusPublished - 2018

Cite this

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title = "Frame properties of systems arising via iterated actions of operators",
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 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.",
author = "Ole Christensen and Marzieh Hasannasabjaldehbakhani",
year = "2018",
doi = "10.1016/j.acha.2018.04.002",
language = "English",
volume = "46",
pages = "664--673",
journal = "Applied and Computational Harmonic Analysis",
issn = "1063-5203",
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Frame properties of systems arising via iterated actions of operators. / Christensen, Ole; Hasannasabjaldehbakhani, Marzieh.

In: Applied and Computational Harmonic Analysis, Vol. 46, No. 3, 2018, p. 664-673.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Frame properties of systems arising via iterated actions of operators

AU - Christensen, Ole

AU - Hasannasabjaldehbakhani, Marzieh

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

U2 - 10.1016/j.acha.2018.04.002

DO - 10.1016/j.acha.2018.04.002

M3 - Journal article

VL - 46

SP - 664

EP - 673

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 3

ER -