Quasi-interpolants Of Bivariate Cubic Spline And Application

Multivariate splines are applied widely in approximation, computer aided geometric design, computational geometry, finite element method and so on. One of the most important questions is how to produce a spline function from given discrete data or a complex function using the B-spline method. Popular methods include interpolation and the least squares approximation. However, both of them require to solve a system of linear equations with as many unknowns as the dimension of the spline space. When the dimension of the spline space becomes bigger, the computing speed is slower. For this purpose local methods, which determine spline coefficients by using only local information, are more suitable. Spline quasi-interpolation is a local method. It need not solve linear equations, so it is efficient. In this paper, we obtain a new quasi-interpolant of bivariate cubic spline.This paper is organized as follows:In chapter1, we briefly introduce the subject of our study, univariate B-spline basis functions and multi-spline functions.In chapter2, we discuss some spline spaces on an uniform type-2triangulation. And we focus on the linear independence of some spline functions with the smallest support. We find a subspace of the bivariate cubic spline, which can be generated by a linearly independent spline functions. We also present some good properties of these spline functions.In chapter3. we introduce some results on an univariate spline quasi-interpolant which can reproduce the whole univariate spline space, and similarly we construct a quasi-interpolant of bivariate cubic spline. We approximate some test functions in other paper by the operator we constructed, and the experimental results demonstrate it has good approximation order and high efficiency.In chapter4. as an application of spline quasi-interpolation, we present an two-dimensional cubature formula and solve two two-dimensional Fredholm integral equations of the second kind. The numerical results show that the results obtained by our formula have a better approximation and higher efficiency than other methods. Besides, the results are also better than those obtained by the tensor product Simpson formula.