Researches On Constraint Interpolation And Spline Quasi-interpolation Problems

In1946, I. J. Schoenberg systematically studied and established the basic theory of univariate spline. From then on, researches and applications on splines become more and more widely. With the deepening of study on spline function, univariate splines are no longer appropriate to solve many practical problems. In1975, R.H. Wang established a new approach to study the basic theory on multivariate spline function on arbitrary partitions using the methods of function theory and algebraic geometry, and presented the so-called the conformality of smoothing cofactor (CSC) method. Fruitful results of multivariate splines have been obtained up to now. In this paper, our research focuses on problems with constrained length and area, an important kind of interpolation problem in geometric modeling, we also present a numerical scheme for differential equation, and make some research on multivariate spline space S53(△mn(2)) by CSC method. We summary the main work as follows:In Chapter1, we present the background of the researches in this thesis.In Chapter2, geometric constraint problem is a basic and important problem in geo-metric modeling, especially in industry design and industry manufacture, we give a discrete method by using rational Bezier curve with de Casteljau algorithm. Implementation algo-rithm and error conclusion are given in the same time. With this method, we also consider the junction problems for curves with constrained conditions. Smooth interpolation problem with function values, constrained length and given curvature is also considered in this chap-ter. Finally, we give the algorithm and error conclusion for problems with constrained area on rotation surface and on rectangle region. Numerical example also shows the validness and efficiency of the proposed method.In Chapter3, spline quasi-interpolation is applied in many fields, and it is also an important content in approximation theory. Radial basis function (RBF) quasi-interpolation is also a common scheme. We present a multi-level univariate quasi-interpolation scheme with better performance by using two quasi-interpolation operator Q4and LR. We obtain a numerical method for KdV equations by applying the quasi-interpolation scheme to these equations, which has many advantage, such as high accuracy. We also give a new manner for obtaining numerical solution of KdV equation at simpler and more efficient conditions, i.e., blending low degree spline quasi-interpolation operators. Compared with multi-level univariate quasi-interpolation scheme, it has a merit of better stability for error controlling.In Chapter4, it is important to make research on multivariate spline for the diversity and complexity of objective things. we study quintic spline space with2-type triangulation using the conformality of smoothing cofactor method and give two methods for obtaining the basis functions. The Fourier transform formulas are also given for these basis functions. Then we give a kind of spline quasi-interpolation operators with higher accuracy by the ob-tained basis functions which can reconstruct the polynomial space P3∪span{x3y, x2y2, xy3}. By applying these quasi-interpolation operators to numerical solution for2D Burgers equa-tion we get a new numerical scheme. Compared to other numerical methods, it has advan-tages of simple construction, easy implementation. Meanwhile, image reconstructions using these quasi-interpolation operators are employed to show validness and efficiency of the corresponding scheme. Finally, we study the non-uniform spline space S53(△mn(2)) with mul-tiple knots, and give the methods for obtaining basis functions with the unit decomposition property and non-uniform B-spline surface with boundary degeneration.