Optimal Error Analysis Of Weighted Integration For Anisotropic Besov-Wiener Spaces

Since some people, such as Traub, Wozniakowski, have been creating information-based complexity theory in the eighties of last century, numerical integration and approximation problem of multivariate functions have been the main research sub-jects in this field. Recently there are a lot of papers which study the computational complexity of multivariate integration and approximation for various function spaces in different settings. For example, Nikolskii space,Sobolev space,Holder space, op-timal numerical algorithms for the integration and approximation problem were obtained, the asymptotic orders of convergence rate of the n-th minimal error are determined. At present, all the anisotropic function spaces attract scholars'atten-tion abroad and home widely. They not only play an important part in mathematical theories, but also are widely used in wavelet analysis and biostatistics. With the previous study methods as a guide, this paper focuses on the integral problems in the anisotropic Besov-Wiener spaces and Besov spaces with mixed norms with the main tool-functional analysis. This paper also works out the optimal numerical al-gorithms and the asymptotic orders of convergence rate of the n-th minimal error from the worst case, stochastic and average case settings with the explaination of convergence rate of these algorithms.This paper have four chapters:Chapter 1 This chapter mainly summarizes the develop course and the current research directions of information-based complexity and narrate the general theory of information-based complexity, it is explained that the n-th minimal error is a crucial method in order to study the complexity of integration problem in different settings, special function spaces and information classes.Chapter 2 This chapter mainly considers the integration problem on multi-variate periodic functions of anisotropic Besov spaces with mixed norms in the worst case, stochastic and average case settings. We first construct a new algorithm using the Dirichlet interpolation operator in order to estimate the upper bound, and use bump function in order to estimate the lower bound. the asymptotic orders of con-vergence rate of the n-th minimal error are established in detail.Chapter 3 This chapter mainly considers the weighted integration problem on multivariate non-periodic functions of anisotropic Besov-Wiener spaces in the worst case, stochastic and average case settings. We construct a new interpolation poly-nomial algorithm in order to estimate the upper bound, and use bump function in order to estimate the lower bound. the asymptotic orders of convergence rate of the n-th minimal error are determined.Chapter 4 This chapter mainly introduces tractability of multivariate prob-lem, and points out the work needing to deal with in the further research.