Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses

摘要

We study Parseval frame wavelets in L~2 (R~d) with matrix dilations of the form (Df)(x) = 2~(1/2)f(Ax), where A is an arbitrary expanding n x n matrix with integer coefficients, such that ∣det A∣ = 2. In our study we use generalized multiresolution analyses (GMRA) (V_j) in L~2(R~d) 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 L~2(R~d). As an application, we include an explicit construction of an orthonormal wavelet on the real line whose dimension function is essentially unbounded.