Construction of symmetric or anti-symmetric B-spline wavelets and their dual wavelets

Abstract

Suppose N _m (x) is the B-spline function of order m. An explicit construction algorithm for the wavelet with symmetry

associated with N _m (x) is presented, where n is an arbitrary positive integer and 4 does not divide (m+n). By appropriately selecting n, we can obtain the B-spline wavelet with short support or arbitrarily high vanishing moments. When 4 does not divide m+1, we prove that ψ _{m, 1}(x) corresponding to N _m (x) has the shortest support among the wavelets whose scaling functions have an approximation of order m. Moreover, the dual scaling function Ñ _m (x) and the dual wavelet ψ˜ _{m, n} (x) are also constructed explicitly. Thereby, Ñ _m (x) and ψ˜ _{m, n} (x) are symmetric or anti-symmetric. Furthermore, we study the regularity of Ñ _m (x). Particularly, we find that as n increases, the order of vanishing moment of ψ _{m, n} (x) as well as the regularity of Ñ _m (x) also increases. Two examples are given to illustrate our results.

Keywords:

• B-spline wavelet
• dual wavelet
• symmetry
• vanishing moment
• regularity

2000 AMS Subject Classifications :

• 42C15
• 94A12

Acknowledgements

The authors thank the two reviewers for their valuable suggestions that improved the presentation of this paper. The authors are supported by the Natural Science Foundation of Guangdong Province (Nos. 05008289, 032038) and the Doctoral Foundation of Guangdong Province (No. 04300917) while writing this paper. This work has been supported by the Natural Science Foundation of Guangdong Province (Nos. 06105648, 05008289, 032038), and the Doctoral Foundation of Guangdong Province (No. 04300917).