Univariate and bivariate orthonormal splines and cardinal splines on compact supports

In this paper, univariate and bivariate orthonormal spline functions with small compact supports are constructed. The properties of orthonormal splines are explored. Furthermore, cardinal spline functions with small compact supports are built. Numerical integration formulas are created as the applications of the cardinal splines. The method can also be used to construct high-dimensional orthonormal splines, which are not cross products of univariate splines and have their own advantages.