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Orthonormal wavelets and shift invariant generalized multiresolution analyses


Authors: Sharon Schaffer Vestal and Eric Weber
Journal: Proc. Amer. Math. Soc. 131 (2003), 3089-3100
MSC (2000): Primary 42C40, 46N99
DOI: https://doi.org/10.1090/S0002-9939-03-06928-4
Published electronically: January 15, 2003
MathSciNet review: 1993218
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Abstract: All wavelets can be associated to a multiresolution-like structure, i.e. an increasing sequence of subspaces of $L^2({\mathbb R})$. We consider the interaction of a wavelet and the shift operator in terms of which of the subspaces in this multiresolution-like structure are invariant under the shift operator. This action defines the notion of the shift invariance property of order $n$. In this paper we show that wavelets of all levels of shift invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral dilation factors.


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Additional Information

Sharon Schaffer Vestal
Affiliation: Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395
Address at time of publication: Department of Computer Science, Mathematics and Physics, Missouri Western State College, St. Joseph, Missouri 64507
Email: sharonv@mwsc.edu

Eric Weber
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Address at time of publication: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-3036
Email: weber@math.tamu.edu, esw@uwyo.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06928-4
Keywords: Wavelet, GMRA, shift invariant space
Received by editor(s): April 23, 2002
Published electronically: January 15, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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