Orthonormal wavelets and shift invariant generalized multiresolution analyses

Vestal, Sharon; Weber, Eric

doi:10.1090/S0002-9939-03-06928-4

Orthonormal wavelets and shift invariant generalized multiresolution analyses
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by Sharon Schaffer Vestal and Eric Weber PDF

Proc. Amer. Math. Soc. 131 (2003), 3089-3100 Request permission

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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