On quadrature error expansions. Part I

摘要

We treat the theory of numerical quadrature over a square using an m2 copy Q(m)ƒ of a one-point quadrature rule. For some integrand functions the quadrature error Q(m)ƒ − Iƒ may be expressed as an asymptotic expansion in inverse powers of m or other simple functions of m. We determine in some cases the nature of this expansion and derive integral representations for both the coefficients and the remainder term. In this part we deal only with smooth functions and those having algebraic line singularities along edges. In some of these cases the form is already known but some of the integral representations are new. These results form the basis for Part II in which new expansions, for integrands having algebraic singularities along intersecting edges and point algebraic singularities at the vertices will be presented.