Existence theorem and minimal cardinality of UEP framelets and MEP bi-framelets (with Z. Zhang), Applied and Computational Harmonic Analysis, vol.34, no.2, pp.297-307, 2013.
Based on multiresolution analysis (MRA) structures combined with the unitary
extension principle (UEP), many frame wavelets were constructed, which are
called UEP framelets. The aim of this letter is to derive general properties
of UEP framelets based on the spectrum of the center space of the underlying
MRA structures. We first give the existence theorem, that is, we give a
necessary and sufficient condition that an MRA structure can derive UEP
framelets. Second, we present a split trick that each mother function can be
split into several functions such that the set consisting of these functions
is still a UEP framelet. Third, we determine the minimal cardinality of UEP
framelets. Finally, we directly construct UEP framelets with the minimal
cardinality. Based on a pair of multiresolution analysis (MRA) structures,
when their spectra intersect, we can always construct a pair of dual frame
wavelets using mixed extension principle (MEP). This pair of dual frame
wavelets are called a pair of MEP bi-framelets. We also give the split trick
and find out the minimal cardinality of such MEP bi-framelets.
UEP framelets, MEP bi-framelets, spectrum, cardinality.
Get the full paper (revised on 08/15/12): PDF file.
Get the official version via doi:10.1016/j.acha.2012.08.005.
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