A novel method of constructing compactly supported orthogonal scaling functions from splines

A novel construction of compactly supported orthogonal scaling functions and wavelets with spline functions is presented in this paper. Let M n be the center B-spline of order n , except for the case of order one, we know M n is not orthogonal. But by the formula of orthonormalization procedure, we can construct an orthogonal scaling function corresponding to M n . However, unlike M n itself, this scaling function no longer has compact support. To induce the orthogonality while keeping the compact support of M n , we put forward a simple, yet efficient construction method that uses the formula of orthonormalization procedure and the weighted average method to construct the two-scale symbol of some compactly supported orthogonal scaling functions.