On Singularity Of The Dimensions Of The Bivariate Spline Spaces Over A Type Of Partition

Multivariate spline function is an important research object in approximation theory, numerical analysis, numerical solutions of differential equations, etc and are widely ap-plied in engineering. A multivariate spline space is a finite dimensional linear space. The dimension of a spline space is important for the theories and applications on multivariate splines. However, the dimension of a spline space is such a difficult problem that we can only obtain the dimensional formulae of some special spline spaces. In general case, the dimension formula is unknown, especially when the spline space is singular. In this thesis, we utilize the smoothing Cofactor-conformality method to discuss the singularities of bi-variate spline spaces on some partitions, obtain some necessary and sufficient conditions to the singularities of these spline spaces.In chapter1, we briefly introduce some basic theories of multivariate splines and the Smoothing Cofactor-Conformality method, which is the basis of our following work. In the following chapters, we use the smoothing Cofactor-Conformality method to research on singularities of bivariate spline spaces over some special partitions.In chapter2, we introduce some basic results of the dimensions of spline spaces. Firstly, we give the dimensional formulae of spline spaces on crosscut partition or quasi-crosscut partitions. Secondly, we give some basic results of triangulations, dimensional formulae of spline spaces on triangulations. Finally, we introduce some results of the singularity of the Morgan-Scott triangulations and the stability of the dimensions of general spline spaces.In chapter3, we use the smoothing Cofactor-conformality method to analyze the spline spaces on a partition of polygon, obtain some properties of these spline spaces which are usefull to research the singularities of bivariate spline spaces over some other partitions. We obtain some necessary and sufficient conditions to the singularities of these spline spaces.