Random variable dilation equation and multidimensional prescale functions
Authors:
Julie Belock and Vladimir Dobric
Journal:
Trans. Amer. Math. Soc. 353 (2001), 47794800
MSC (2000):
Primary 60A10, 60G50; Secondary 42C40, 42C15
Published electronically:
June 21, 2001
MathSciNet review:
1852082
Fulltext PDF Free Access
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Abstract: 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:
http://dx.doi.org/10.1090/S0002994701028331
PII:
S 00029947(01)028331
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
