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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Random variable dilation equation and multidimensional prescale functions


Authors: Julie Belock and Vladimir Dobric
Journal: Trans. Amer. Math. Soc. 353 (2001), 4779-4800
MSC (2000): Primary 60A10, 60G50; Secondary 42C40, 42C15
Published electronically: June 21, 2001
MathSciNet review: 1852082
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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

\begin{displaymath}\varphi \left( x\right) =\left\vert \det M\right\vert \sum_{k\in \Gamma} c_{k}\varphi \left( Mx-k\right) \text{.} \end{displaymath}

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.


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Additional Information

Julie Belock
Affiliation: Department of Mathematics, West Chester University of Pennsylvania, West Chester, Pennsylvania 19383
Address at time of publication: Department of Mathematics, Salem State College, Salem, Massachusetts 01970
Email: jbelock@salemstate.edu

Vladimir Dobric
Affiliation: Department of Mathematics, Lehigh University, 14 Packer Avenue, Bethlehem, Pennsylvania 18015
Email: vd00@lehigh.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02833-1
PII: S 0002-9947(01)02833-1
Keywords: Dilation equations, tilings, wavelets
Received by editor(s): January 10, 2000
Received by editor(s) in revised form: January 8, 2001
Published electronically: June 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society