Multiscaling frame multiresolution analysis and associated wavelet frames

Abstract

Frame multiresolution analysis (FMRA) in [Formula: see text] is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in [Formula: see text], which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity [Formula: see text]. We characterize wavelet frames and Parseval wavelet frames for [Formula: see text] under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions [Formula: see text] in [Formula: see text] to be multiframe generators of [Formula: see text]. Especially, the second condition depends on the multiscaling FMRA and [Formula: see text] only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.