Existance theorem and minimal cardinality of UEP framelets and MEP bi-framelets

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.