On Box Spline Computation And Multivariate Polynomial Interpolation

Box spline functions and Birkhoff interpolation both have wide applications in nu-merical analysis and approximation theory. As generalizations of B-splines and univariate truncated powers respectively, Box splines and multivariate truncated powers have close re-lationship with partition function, and play important roles in many fields such as discrete mathematics and combinatorics. Birkhoff interpolation is a kind of interpolation problem which allows inconsecutive derivative conditions on points, and it attracts more and more attentions since it is a general case of Lagrange interpolation and Hermite interpolation; principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation. Both Birkhoff interpolation and principal lattice are of great im-portance in computer aid geometric design and finite element method. This dissertation mainly studies computation of explicit expression of Box splines and multivariate trun-cated powers, smoothness degree of Box splines in each direction of partition edges, and two-dimensional Birkhoff interpolation problems and error analysis of principal lattice in-terpolation. The main contributions include:1. Computing explicit expression of Box splines and multivariate truncated powers is a very difficult problem. Traditional methods are only used to compute them with specific in-cidence matrix, without giving general explicit expressions. [19] pointed out that the Laplace transform of truncated powers can be regarded as an expression of hyperplane arrangements which is a finite set of affine hyperplanes in a finite dimensional vector space. Based on this theory, we compute explicit expressions of Box splines and multivariate truncated powers associated with a matrix containing multi-column vectors in two- and three-dimensional spaces by the reduced theorem in hyperplane arrangements, and final obtain the general explicit expressions.2. Box splines are spline functions whose partition edges are the linear shifts of col-umn vectors in their incidence matrix, and their smoothness degrees on each direction of partition edges are unknown. It is important to compute the smoothness degree for better understanding properties of Box splines. By using the Conformality of Smoothing Cofactor method, we compute the least smoothness degree on each direction of partition edges about Box splines and multivariate truncated powers in two-dimensional spaces by using our pre- vious work regarding the explicit formulas. This result improves the previous results about the smoothness degree of Box splines, and gives a better understanding on structures of Box splines and multivariate truncated powers.3. Birkhoff interpolation is an interpolation problem satisfying inconsecutive derivative conditions. As a general case of Lagrange interpolation and Hermite interpolation, the study of Birkhoff interpolation is so difficult that the unisolvent condition is not always satisfied even in univariate cases. Constructing Birkhoff interpolation functional set is a useful tool to solve Birkhoff interpolation problems. By using the so-called superposition straight line method in multivariate polynomial interpolation proposed by Liang, we introduce Birkhoff interpolation functional set along algebraic curve into Birkhoff interpolation, and propose a superposition method of constructing a bivariate Birkhoff interpolation functional set. Our method chooses the interpolating nodes and their derivative directions in a free way, and hence solves the Birkhoff interpolation problem effectively.4. Principal lattice is a classical simplicial configuration of suitable nodes for multivari-ate polynomial interpolation, which consists of intersection points of parallel hyperplanes. Principal lattice is widely applied in finite element methods, and the associated error anal-ysis is an important topic. With the help of special structures of principal lattices, we compute errors of a kind of regular principal lattices:generalized principal lattices. The results show that the error of generalized principal lattices of degree m depends on the fol-lowing factors:the range of interpolation area, the m+1-th order derivative of interpolation function, and distance proportions among hyperplanes in the same pencil. That makes us understand more clearly about intrinsic factors which produce errors of generalized principal lattices. Moreover, we give a more precise method for computing upper bounds of errors on generalized principal lattices. Numerical experiments show that the upper bound of errors on generalized principal lattices of low degree obtained by our method is more precise than the results obtained by previous methods.