Abstract
We resolve a long-standing question on Lp 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 Lq(ℝd) 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 L2. Completeness of the Mexican hat and other wavelet systems in Lp 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