Random variable dilation equation and multidimensional prescale functions

Abstract

Abstract. A random variable Z satisfying the random variable dilation equation MZ d = Z + G, where G is a discrete random variable independent of Z with values in a lattice Γ ⊂ R d and weights {c k } k∈Γ and M is an expanding and Γ-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density ϕ which will satisfy a dilation equationWe have obtained necessary and sufficient conditions for the existence of the density ϕ and a simple sufficient condition for ϕ's existence in terms of the weights {c k } k∈Γ . 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 ϕ. The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of ϕ allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when ϕ is a prescale function.

Low-Pass Filters and Scaling Functions for Multivariable Wavelets

“…where D is a digit set for A. The existence of such tilings has been studied by Lagarias and Wang [9][10][11][12]), He and Lau [7], Belock and Dobric [3], and the author [1,2]. This paper investigates general multivariable MRAs.…”

A pseudo-probabilistic approach to the dilation equation for wavelets

“…We call M Γ = Γ even the evens, because on R when Γ = Z, it is exactly the even integers, and we call the other coset Γ odd the odds, for the analogous reason. Then the condition for µ to be absolutely continuous is that k∈Γeven p k = k∈Γ odd p k = 1/2 (as originally proved in [2]). The significance of this condition is explained by the following result.…”

“…If p k ≥ 0 for all k, then µ 1 will be a probability measure (rather than just a pseudo-probability measure). This is the case considered by Belock and Dobric [2]. In this case, every µ n will be a probability measure, so trivially we have µ n TV = 1 for all n, giving a solution measure µ which is also a probability measure.…”

“…Depending on conditions placed upon {p k }, a functional solution can be a scaling function (meaning that its translates form an orthonormal basis in the MRA) or merely a prescale function (meaning that its translates form a Riesz basis). Dobric, Gundy, and Hitczenko (in the one-dimensional case) [6] and Belock and Dobric (in the multidimensional case) [2] considered a probabilistic approach to equation (1). This is natural because, under the condition that all p γ are non-negative, the right-hand side of the dilation equation can be interpreted as the convolution of two probability measures.…”

“…In particular, every real number can be expressed an an integer plus a binary expansion of the above form. Similarly, every point in R 2 can be expressed as a Gaussian integer plus a binary expansion in the sense of equation (2).…”