Construction of symmetric or anti-symmetric B-spline wavelets and their dual wavelets
Youfa Li, Shouzhi Yang · International Journal of Computer Mathematics · 2011
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.