SOLUTION OF 2 PROBLEMS ON WAVELETS

We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space H-2(R). The second is the existence, given any regular wavelet basis for L(2)(R), of a Multi-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for L(2)(R(n)) arises from a Multi-Resolution Analysis that is regular module the action of a unitary operator, which is when n = 2 a Calderon-Zygmund operator of convolution type.