Construction of Spline Type Orthogonal Scaling Functions and Wavelets

In this paper, we present a method to construct orthogonal spline-type scaling functions by using B-spline functions. B-splines have many useful properties such as compactly supported and refinable properties. However, except for the case of order one, B-splines of order greater than one are not orthogonal. To induce the orthogonality while keeping the above properties of B-splines, we multiply a class of polynomial function factors to the masks of the B-splines so that they become the masks of a spline-type orthogonal compactly-supported and refinable scaling functions in L2. In this paper we establish the existence of this class of polynomial factors and their construction. Hence, the corresponding spline-type wavelets and the decomposition and reconstruction formulas for their Multiresolution Analysis (MRA) are obtained accordingly