Optimal Approximation And Optimal Quadrature Problems

Width theory and optimal algorithm theory are two important research directions in function approximation theory.They have both theoretical significance and practical application prospect.In this paper,we focus on the problem of the optimal quadrature formula,which is one of the research directions of the optimal algorithm theory,that is,the problem of constructing the optimal approximation algorithm for the definite integral.Since some smooth functions can be represented as convolutions with cyclic variation diminishing kernels,that does not apply to some classes of analytic functions such as the functions in periodic Hardy-Sobolev classes.We hope,based on the idea of constructing a general theory covering smooth and analytic cases,the properties of the new class of functions and the expressions of their optimal quadrature formulas are studied by using the viewpoint and method of functional analysis,and the accurate errors are expected to be calculated.In this paper,we consider the optimal quadrature problem on classes of functions defined by operators with certain oscillation properties.We study the error estimate of the optimal quadrature formula,which is divided into two parts:first,we discuss the error upper bound estimation Lemma,And then,by discussing the solution of its minimum norm problem and using modern mathematical tools as Borsuk’s theorem,the lower bound of the error is estimated.Thus,the result of the optimal quadrature formulae about information composed of the values of a function and its kth(k-1,…,r-1)derivatives on free knots for the classes (?) are obtained,and the error estimates of the optimal quadrature formulae are exactly determined as follows:Let n,N∈N,r ≥ 2,and[x]be the greatest integer no bigger than x.Then the rectangular quadrature formula (?) is optimal for the class of functions (?) among all quadrature formulae of the form where the nodes 0≤t1≤…≤tn<2π and the coefficients ak∈R are arbitrary,vk(k=1,…,n)are positive integers satisfying the condition:(?).Moreover,the error of the optimal quadrature formulae on the class of functions (?) is(?)...