Characterizations of orthonormal scale functions: a probabilistic approach

The construction of a multiresolution analysis starts with specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of $L_2(R)$. Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in the case where the scale function is continuous. These results extend and clarify those of A. Cohen, and Hernandez, Wang, and Weiss. The method also applies to more general dilation schemes that commute with translations by $Z^d$.