### Abstract

We resolve a long-standing question on L^{p} completeness of the time-scale (or wavelet) system generated by the Mexican hat function, when p≥2. Our main result concerns frequency-scale systems generated by modulation and dilation of a single function. The mixed frame operator (analysis followed by synthesis) is shown to be bijective from L^{q}(ℝ^{d}) to itself, for 1≤q<∞, so that the frequency-scale synthesis operator is surjective. Tools include the discrete Calderón condition and a generalization of the Daubechies frame criterion in L^{2}. Completeness of the Mexican hat and other wavelet systems in L^{p} then follows for 2≤p<∞, by Fourier imbedding of frequency-scale systems.

Original language | English (US) |
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Pages (from-to) | 163-189 |

Number of pages | 27 |

Journal | Constructive Approximation |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2011 |

### Keywords

- Analysis
- Completeness
- Fourier space
- Spanning
- Synthesis
- Wavelet

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

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## Cite this

*Constructive Approximation*,

*33*(2), 163-189. https://doi.org/10.1007/s00365-010-9098-3