Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses

AbstractWe study Parseval frame wavelets in L2(Rd) with matrix dilations of the form (Df)(x)=2f(Ax), where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. In our study we use generalized multiresolution analyses (GMRA) (Vj) in L2(Rd) with dilations D. We describe, in terms of the underlying multiresolution structure, all GMRA Parseval frame wavelets and, a posteriori, all semi-orthogonal Parseval frame wavelets in L2(Rd). As an application, we include an explicit construction of an orthonormal wavelet on the real line whose dimension function is essentially unbounded