The Improvement Of The Functional Integral Error In The Reproducing Kernel Hilbert Space

Multivariate integration and approximation is a major research problem in computational mathematics. Today, more and more people pay close attention to it and it appears in many papers. Furthermore, it is applied in lots of fields including physical, chemistry, financial, engineering and so on.Besides, we know that function integral error is one of the problems of interest among researchers, the results of which are applied in other multivariate problems. In large quantity practical problems, we need to deal with multivariate functions. Then the traditional single variable function integral analysis methodology of approximation error does not work any all. (In traditional analysis methodology, the number of variates d is fixed, while now the number of variates d is very large). Recently, in the branches of the IBC(Information Based Complexity)theory, more and more problems about tractability of multivariate problems arise. So, in the research of tractability, people pay more attention to weighed function spaces than isotropic or anisotropic function space. Professor I.H.Sloan and professor H.Wozniakowski studied tractability and strong tractability firstly in weighed spaces Fd, and mainly studied multivariate integration in reproducing kernel Hilbert space Fd.The improvement of multivariate integral error is studied in this paper. It mainly focuses on two settings. One is the worst case setting; and the other is the average case setting. In this paper, we consider I(f)=∫f(x)dx is approximated by algorithm U(f)=(?)vif(xi).Then error appears. When we choose the optimal weighed and nodes, the integral error can be improved. Hence it is significant to study the improvement of multivariate integral error. There are three parts in this paper.In the first chapter,the background of tractability and multivariate integration is introduced and some basic theories about functional used in the last two chapters are presented, including linear space, normed linear space, inner product, reproducing kernel, reproducing kernel Hilbert space, variance, covariance and so on. At the same time, the questions which will be studied in the last chapters are simply introduced. In the second chapter,mainly studied the improvement of the multivariate integral error in reproducing kernel Hilbert space, using the rule of integration U(f)=(?)vif(xi) to approximate multivariate integration, by constructing functions,we get the upper bound of integral error: And we give some simple examples in order to illustrate the theory provided in this paper. In the third chapter, mainly studied the improvement of the multivariate integral error about I(f)=∫xf(x)dx in reproducing kernel Hilbert space in the average case. According to two important lemmas, we obtain the conclusion of this chapter:In order to explain the theorem given in this chapter, we also bring in some examples.