Construction of a family of non-stationary biorthogonal wavelets

• Published in: Journal of inequalities and applications 2019-11, Vol.2019 (1), p.1-15, Article 286
• Main Authors: Zhang, Baoxing, Zheng, Hongchan, Zhou, Jie, Pan, Lulu
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; B spline functions; Coifman biorthogonal wavelets; Exponential pseudo-splines; Masks; Mathematics; Mathematics and Statistics; Multiresolution analysis; Pseudo-splines; Stability analysis; Vanishing moments; Wavelet analysis
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-019-2240-2

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.