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.
Nguyen, Tung, "Construction of Spline Type Orthogonal Scaling Functions and Wavelets" (2015). Honors Projects. 19.