Random variable dilation equation and multidimensional prescale functions

Abstract:

A random variable $Z$ satisfying the random variable dilation equation $MZ \overset {d}{=}Z+G$, where $G$ is a discrete random variable independent of $Z$ with values in a lattice $\Gamma \subset$ $\mathbf {R}^{d}$ and weights $\left \{ c_{k}\right \} _{k\in \Gamma }$ and $M$ is an expanding and $\Gamma$-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density $\varphi$ which will satisfy a dilation equation \[ \varphi \left ( x\right ) =\left | \det M\right | \sum _{k\in \Gamma } c_{k}\varphi \left ( Mx-k\right ) \text {.} \] We have obtained necessary and sufficient conditions for the existence of the density $\varphi$ and a simple sufficient condition for $\varphi$’s existence in terms of the weights $\left \{ c_{k}\right \} _{k\in \Gamma }.$ Wavelets in $\mathbf {R}^{d}$ can be generated in several ways. One is through a multiresolution analysis of $L^{2}\left ( \mathbf {R}^{d}\right )$ generated by a compactly supported prescale function $\varphi$. 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 $\varphi$ 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 $\varphi$ is a prescale function.