Research On Spline Interpolation Poisedness And Interpolation Approach Problems

In science computations, interpolation method play a important role in lots of aspects which are involved in function approximation, multi-statistics, system control and computer aided geometrical design. Therefore, the study concerning theory and applying on function interpolation has been paid more attention. In this thesis, we focus on the study about spline interpolation problem. Here we state our researches as follows briefly.The poisedness of interpolation problem for univariate spline spaces means that for given points t1···,tm in the domain of the spline space and arbitrary m real numbers y1···,ym, if there exist only one spline s in this space such that s(ti) = yi, i=1,···,m.Schoenberg-Whitney theorem [22] provided a condition about determining if the set of interpolation points is a poisedness set. But, there is not any conclusion for the distributaries law of this kind of points sets. It is important to further study to find the essential characters in order to applying easily.(1) The construction laws of poised interpolation set of uni-variable splinesWe introduce following concepts:the local poised sets, the minimal local poised sets, the minimal poised sets and the perfect local poised sets, present and prove some new sufficient and necessary condition for determine the poisedness of interpolation sets, and obtain following results:For univariate spline spaces, the poised set consists of finite perfect local poised sets. These sets were desperate by n-1 spline knots one by one. Any perfect local poised set is constructed by trivial extending finite times from a minimal local poised set. And the minimal local poised set is constructed by trivial extending from a minimal poised set.The construction way of the minimal poised sets is only depended on the order of the spline space and there are only finite configuration schemes.We also provided the algorithms for finding the configuration of the minimalpoised sets and minimal local poised sets.These results can be use to determine the poisedness of a interpolation set for the spline space, and to design or modify the spline curve by choosing spline knots in terms of the given interpolation sets.(2) Configuration of poised interpolation set of bivariate linear splinesWe introduce a new concept so-called exact interpolation set, construct direct tree with a root, and then define weight of vertexes and the cost functions and current functions of the arcus. A sufficient and necessary condition about the configuration law of poised set of the spline interpolation have been obtained. That is:For bivariate spline space, its interpolation set is a poised one if and only if the interpolation set is the union of finite perfect local poised interpolation sets. and the closure of the union of these normal triangle cells which contain some points of A consists of finite connect polygonal domains, which were segregated by some domains, and anyone of these domain consists of some strongly connect normal triangle cells which contain none of net points of A.(3) Quasi-interpolator with interpolation propertyWe present a method to construct a quasi-interpolation operator with certain interpolation property, and prove that the convergence of the sequence consisting of this kind of operators.(4) Random interpolation and random splineWe analyze the problem of random force to thin beam of a structure, introduce the concept of random splines, and study the poisedness of random spline interpolation and the approximation problem in probability convergence.