### Abstract

We consider Gabor frames {e

^{2πibm}·g(· − ak)}_{m,k∈Z}with translation parameter a = L/2, modulation parameter b ∈ (0, 2/L) and a window function g ∈ C^{n}(R) supported on [x_{0}, x_{0}+L] and non-zero on (x_{0}, x_{0}+L) for L > 0 and x_{0}∈ R. The set of all dual windows h ∈ L^{2}(R) with sufficiently small support is parametrized by 1-periodic measurable functions z. Each dual window h is given explicitly in terms of the function z in such a way that desirable properties (e.g., symmetry, boundedness and smoothness) of h are directly linked to z. We derive easily verifiable conditions on the function z that guarantee, in fact, characterize, compactly supported dual windows h with the same smoothness, i.e., h ∈ C^{n}(R). The construction of dual windows is valid for all values of the smoothness index n ∈ Z_{≥0}∪ {∞} and for all values of the modulation parameter b < 2/L; since a = L/2, this allows for arbitrarily small redundancy (ab)^{−1}> 1. We show that the smoothness of h is optimal, i.e., if g /∈ C^{n+1}(R) then, in general, a dual window h in C^{n+1}(R) does not exist.Original language | English |
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Article number | 105304 |

Journal | Journal of Approximation Theory |

Volume | 249 |

Number of pages | 27 |

ISSN | 0021-9045 |

DOIs | |

Publication status | Published - 2020 |

### Keywords

- Dual frame
- Dual window
- Gabor frame
- Optimal smoothness
- Redundancy 2010 MSC
- Primary 42C15
- Secondary 42A60