MRA parseval frame multiwavelets in L^2(R^d)

In this paper‎, ‎we characterize multiresolution analysis (MRA)‎ ‎Parseval frame multiwavelets in $L^2(R^d)$ with matrix‎ ‎dilations of the form $(D f )(x) = \sqrt{2}f (Ax)$‎, ‎where $A$ is an arbitrary expanding $d\times d$ matrix with integer coefficients‎, ‎such that $|detA| =2$‎. ‎We study a class of generalized low pass‎ ‎matrix filters that allow us to define (and construct) the‎ ‎subclass of MRA tight frame multiwavelets‎. ‎This leads us to an‎ ‎associated class of generalized scaling functions that are not‎ ‎necessarily obtained from a multiresolution analysis‎. ‎We also investigate‎ ‎several properties of these classes of generalized‎ ‎multiwavelets‎, ‎scaling functions‎, ‎matrix filters and give‎ ‎some characterizations about them‎. ‎Finally‎, ‎we describe the‎ ‎matrix multipliers classes associated with Parseval frame multiwavelets (PFMWs)‎ ‎in $L^2(R^d)$ and give an example to support our theory‎.