### Abstract

The purpose of the paper is to analyze frames (Formula presented.) having the form (Formula presented.) for some linear operator (Formula presented.). A key result characterizes boundedness of the operator T in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation (Formula presented.) can be achieved for an operator T that has an extension to a bounded bijective operator (Formula presented.) In this case we also characterize all the dual frames that are representable in terms of iterations of an operator V; in particular we prove that the only possible operator is (Formula presented.) Finally, we consider stability of the representation (Formula presented.) rather surprisingly, it turns out that the possibility to represent a frame on this form is sensitive towards some of the classical perturbation conditions in frame theory. Various ways of avoiding this problem will be discussed. Throughout the paper the results will be connected with the operators and function systems appearing in applied harmonic analysis, as well as with general group representations.

Original language | English |
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Journal | Integral Equations and Operator Theory |

Volume | 88 |

Issue number | 4 |

Pages (from-to) | 483-499 |

ISSN | 0378-620X |

DOIs | |

Publication status | Published - 2017 |

### Keywords

- Analysis
- Algebra and Number Theory
- Frames
- Gabor frames
- Operator representation of frames
- Perturbation theory
- 42C15
- math.FA

## Cite this

Christensen, O., & Hasannasab, M. (2017). Operator Representations of Frames: Boundedness, Duality, and Stability.

*Integral Equations and Operator Theory*,*88*(4), 483-499. https://doi.org/10.1007/s00020-017-2370-1