Construction of a family of non-stationary biorthogonal wavelets

Baoxing Zhang,Jie Zhou,Hongchan Zheng,Lulu Pan

doi:10.1186/s13660-019-2240-2

Abstract

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.

Highlights

During the last decades, biorthogonal wavelets have been proved to be very successful tools in engineering and applied mathematics
In this paper, inspired by the abovementioned work, we mainly focus on this subclass of exponential pseudo-splines and construct a family of non-stationary biorthogonal wavelets
5 Conclusion This paper presented a family of non-stationary biorthogonal wavelets. This family of biorthogonal wavelets is based on a subclass of the exponential pseudo-splines, which consists of interpolatory cardinal functions

Summary

Introduction

Biorthogonal wavelets have been proved to be very successful tools in engineering and applied mathematics. To obtain the desired biorthogonal wavelets, we first derive the explicit form of the symbols of the dual refinable functions for this subclass of exponential pseudo-splines. The corresponding biorthogonal wavelets can be obtained by the non-stationary MRAs. For the new refinable and wavelet functions, we show their stability and that both of them have the exponential vanishing moments.

Results

Conclusion