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

Suppose N(m)(x) is the B-spline function of order m. An explicit construction algorithm for the wavelet with symmetry psi(m,n)(x) := 2(-n+1)Sigma(n)(l=0)(-1)(l) n!/l!(n - l)! N(m)(2x - l) 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 psi(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 (N) over tilde (m)(x) and the dual wavelet. (psi) over bar (m,n)(x) are also constructed explicitly. Thereby, (N) over tilde (m)(x) and (psi) over bar (m,n)(x) are symmetric or anti-symmetric. Furthermore, we study the regularity of (N) over tilde (m)(x). Particularly, we find that as n increases, the order of vanishing moment of (psi) over bar (m,n)(x) as well as the regularity of (N) over tilde (m)(x) also increases. Two examples are given to illustrate our results.