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

DOI:
https://doi.org/10.1090/S0002-9947-01-02833-1

Published electronically:
June 21, 2001

MathSciNet review:
1852082

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Abstract | References | Similar Articles | Additional Information

A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation

We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . 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 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 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:
https://doi.org/10.1090/S0002-9947-01-02833-1

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