### Abstract

A frame in a Hilbert space H is a countable collection of elements in H that allows each f Ïµ H to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.

Original language | English |
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Title of host publication | Operator Theory: Advances and Applications |

Volume | 268 |

Publisher | Springer |

Publication date | 2018 |

Pages | 155-165 |

ISBN (Print) | 9783319759968 |

DOIs | |

Publication status | Published - 2018 |

Series | Operator Theory |
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Volume | 268 |

ISSN | 0255-0156 |

### Keywords

- Frames
- Dual frames
- Dynamical sampling
- Operator theory

## Cite this

Christensen, O., & Hasannasab, M. (2018). Frames, operator representations, and open problems. In

*Operator Theory: Advances and Applications*(Vol. 268, pp. 155-165). Springer. Operator Theory, Vol.. 268 https://doi.org/10.1007/978-3-319-75996-8_8