Abstract
The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.
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Zhang, Z.H. Local analysis, cardinality, and split trick of quasi-biorthogonal frame wavelets. Acta. Math. Sin.-English Ser. 27, 203–218 (2011). https://doi.org/10.1007/s10114-011-8172-5
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DOI: https://doi.org/10.1007/s10114-011-8172-5