Frequency-scale frames and the solution of the Mexican hat problem

We resolve a long-standing question on L completeness of the timescale (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(R) 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. Completeness of the Mexican hat and other wavelet systems in L then follows for 2 ≤ p <∞, by Fourier imbedding of frequency-scale systems.