Random variable dilation equations in R(d) and the stability of scaling functions.

摘要

Let Γ be a lattice in Rd and let M be an expansive matrix, meaning all eigenvalues have moduli greater than 1, such that $MG\subseteq G$. A dilation equation has the form $4x=Sg\in Gag4Mx+g,$ where a:= $agg\in G$ is in l1 (Γ). A function which solves a dilation equation is known as a scaling function. If an L2-solution to this equation exists, it may be used to generate multiresolution analyses and wavelet bases, which are powerful tools in signal and image processing [4], [19], [20]. Other applications of scaling functions include interpolating subdivision schemes for computer-aided design [5], [3]. If $a\in l2$ (Γ) then the dilation equation will have a compactly supported solution, which may be a distribution [7], [4]. The problem of finding a solution which is a function can be recast in terms of probability: suppose G₁, G₂, …are i.i.d. random variables on a probability space $W,F,P$ and suppose $Z:=k=1>\infty M-k\; Gk$ is finite. Then Z is a solution to a random variable dilation equation $MZ=dZ+G,$ where Z and G are independent and $G=dG1.$ Gundy [11] has shown that £ (Z) is absolutely continuous if and only if the fractional part of Z is uniform for the one-dimensional case. In this thesis we generalize Gundy's result to higher dimensions under the assumption that G has values in a finite subset of the lattice and prove a sufficient condition on the distribution of G to obtain a solution which has a density function. The techniques used to achieve these results reflect the increased complexity of the problem in higher dimensions, which is due primarily to the fact that M may produce a rotation as well as a dilation, and may even be non-normal. We also present an improved version of an algorithm described by Lawton, Lee and Shen [16] for determining the stability of the scaling function.