Some Problems In The Bivariate Spline Functions

Researches on the spline functions have been motivated by the aim to develop a powerful tool for many applications,such as scattered data fitting,finite element method,constructing surfaces,etc.In view of the variety and complexity of the objectives,it is important to study the bivariate splines.In 1975,Renhong Wang presented the smoothing cofactor conformality method and established a new approach to study the basic theory of multivariate spline function on arbitrary partitions.In this thesis,we discuss the related theory of bivariate spline functions.The main results of this thesis can be summarized as follows:1.In Chapter 1,some background information about bivariate splines,local refinable s-plines,and the stability of the spline basis are introduced.2.In Chapter 2,by using the smoothing cofactor conformality method,we consider the dimension of the bivariate spline spaces on triangulations.The dimension of Sk2(?MS)(k>4)and the geometric condition of S42(?MS)with unstable dimension are discussed,where ?MS is the Morgan-Scott triangulation.If the degree of each interior vertex of the non-degenerate triangulation is at least 6,then the dimension of S2rr(?)(r ? 1)is stable.And we give an example to show that the non-degenerate condition is necessary.3.In Chapter 3,we consider the stability with respect to the Lp(1 ? p<?)norm of the truncated hierarchical B-spline basis.The strong stability with respect to the L? norm of the truncated hierarchical B-spline basis has been investigated by Giannelli et al.,whether the stability with respect to the Lp(1 ? p<?)norm is not clear.The truncated hierarchical B-spline basis is weakly stable with respect to the Lp(1 ? p<?)norm.This means that the associated constants to be considered in the stability analysis depend on the number of the hierarchy depth.4.In Chapter 4,we extend the truncated hierarchical B-splines to the bivariate splines on arbitrary partitions since the splines on the triangulations have the advantage of flexibility and lower degree with the same continuity comparing to tensor-product splines.In addition,we consider a kind of hierarchical quasi-interpolation operators and the numerical examples show that the quasi-interpolation have good approximation order and lower degree of freedom.