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 [Formula: see text] be the center B-spline of order n, except for the case of order one, we know [Formula: see text] is not orthogonal. But by the formula of orthonormalization procedure, we can construct an orthogonal scaling function corresponding to [Formula: see text] . However, unlike [Formula: see text] itself, this scaling function no longer has compact support. To induce the orthogonality while keeping the compact support of [Formula: see text] , 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.