On Parseval Wavelet Frames via Multiresolution Analyses in

We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

Construction of a family of non-stationary biorthogonal wavelets

The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D–D interpolatory subdivision.