The aim of this research is to contribute both the theory and applications of multivariate splines. The theoretical questions to be addressed involve dimensions of multivariate spline spaces, construction of local bases for these spaces, and study of their approximation power. In addition to the traditional spaces defined on triangulations, new spaces defined on triangulations of the sphere or other 3D objects will also be investigated. The applications involve the use of simulated annealing to construct optimal triangulations, the design of shape preserving interpolation and data fitting methods, the use of radial basis functions for penalized least squares fitting, and adaptive methods for fitting functions of many variables.
