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Abstract
We consider a generalization of the notion of non-uniform multiresolution analysis (NUMRA) which is called vector-valued non-uniform multiresolution analysis (VNUMRA). The concept of NUMRA was introduced by Gabardo and Nashed based on the theory of spectral pairs. Xia and Suter introduced vector-valued multiresolution analysis and orthogonal vector-valued wavelets. We introduce VNUMRA where the associated subspace of
has, an orthonormal basis, a collection of translates of a vector-valued function
of the form
where
,
is an integer and
is an odd integer with
such that
and
are relatively prime and
is the set of all integers and the corresponding dilation factor is
. We obtain the necessary and sufficient condition for the existence of associated wavelets and present a construction of VNUMRA.
Acknowledgments
The second author was supported by the UGC Project.