High Dimensional Spline Approximation With A Derivative

Come nearly twenty years, approximation theory of width theory has made great progress, so far, has formed a set of relatively complete, with a considerable range of abstract space point set width theory, has done some analysis has basic significance function in certain scale width estimation, including some very detailed accurate estimates, and to solve the problem, approximation theory methods and skills to get new development.This paper is mainly to solve the weighted cycle Besov approximation problem, cycle approximation problem is a classical approximation of functions in modern mathematical calculation of practical application of a new topic, common ground says is calculated using the known functions to some specific high order derivative properties of approximation degree. Due to the classic Dirichlet and Vallee-poussin nucleus in solve the weighted problem is not suitable, so we hope to classical function transformation, basic keep the classic calculation thought, fusion analysis of Fourier transform method, the new class of functions to do order estimation. At the same time, the discretization method can’t give a specific function approximation space, in this paper; we give the spline wavelet space specific function characterizations. By defining a new discrete function, get new representation theorem:Use Dirichlet and Vallee-poussin nuclear, The results are as follows:Make k∈N, r>1and1<p<2, So there isFurther, by use of spline wavelet as the approximation space, the weighted Besov function class the following characterization. If f∈Lp(Rd)then the...