Frequency-Scale Frames and the Solution of the Mexican Hat Problem

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 L². Completeness of the Mexican hat and other wavelet systems in L^p then follows for 2≤p<∞, by Fourier imbedding of frequency-scale systems.