Biorthogonal spline type wavelets

Let φ be an orthonormal scaling function with approximation degree p − 1, and let Bnbe the B-spline of order n. Then, spline type scaling functions defined by ƒn = ƒ * Bn (n = 1, 2, ...) possess higher approximation order, p + n − 1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended to the case of biorthogonal scaling function system. As an application of the method supplied in this paper, one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function. In particular, if both the method and the lifting scheme of Sweldens (see [1]) are applied, then all pairs of biorthogonal spline type scaling functions shown in references [2] and [3] can be constructed from the Haar scaling function.