Integration Error Estimates Of Periodic Functions In Anisotropic Besov Spaces

The problem of multivariate numerical integration has its applications in many fields, including physics, chemistry, finance, economy, and computational science. This problem is almost impossible to be solved by analytical methods, but only to be solved by numer-ical approximation methods and satisfing the error is no more thanε. Simply speaking, the complexity of multivariate numerical integration is the minimal cost which is needed to solve the d-dimensional multivariate numerical integration problem when the error is no more thanε, which is closely related to the problem is the nth minimal error. In the paper, we studied integration error estimates of periodic functions in anisotropic Besov space, the periodic functions in anisotropic Besov space is that the function of each vari-able has different differentiable properties. It is matched with many models in practical application and there are many applications in numerical integration, such as function approximation, wavelet analysis, differential and integral equations, probability of partial differential equations.Recently, there are a lot of papers which study the integration and the approxi-mation for anisotropic function spaces and isotropic function spaces in different settings. Temlyakov studied integration error estimates of periodic functions in anisotropic Sobolev class and Nikolskii class of function in the worst case setting. In the paper, we mainly study integration error estimates of periodic functions in anisotropic Besov class in the worst setting and the randomized setting and obtain the asymptotic order of convergence rate for the nth minimal error upper bounds and lower bounds.In this paper, it is mainly describe the current resear directins of information com-plexity, and give the basic definitions and basic theories which is useful to study the second chapter. And then describ the mainly research results of information-based complexity, and consider integration errors of periodic functions of anisotropic Besov spaces in the worst setting and the randomized setting, study the upper bounds and lower bounds for the nth minimal error of multivariate integration problem, prov the numerical integra-ton complixity, give the asymptotic order of convergence rate for the nth minimal error bounds of quaduature formala, and get the results of errors in the worst setting and the randomized setting.At last, summarize up the research results of integration error analysis, and propose future direction to study.