Random variable dilation equation and multidimensional prescale functions

A random variable Z satisfying the random variable dilation equation MZ (d) double under bar= Z + G, where G is a discrete random variable independent of Z with values in a lattice Gamma subset of R-d and weights {c(k)}(k is an element of Gamma) and M is an expanding and Gamma -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a densit y phi which will satisfy a dilation equation [GRAPHIC] We have obtained necessary and sufficient conditions for the existence of t he density phi and a simple sufficient condition for phi 's existence in te rms of the weights {c(k)}(k is an element of Gamma). Wavelets in R-d can be generated in several ways. One is through a multiresolution analysis of L- 2 (R-d) generated by a compactly supported prescale function phi. The presc ale function will satisfy a dilation equation and its lattice translates wi ll form a Riesz basis for the closed linear span of the translates. The suf ficient condition for the existence of phi allows a tractable method for de signing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient con dition is necessary in the case when phi is a prescale function.