Modified product cubature formulae

In the univariate case, there is a well-developed theory on the error estimation of the quadrature formulae for integrands from the Sobolev classes of functions. It is based on the Peano kernel representation of linear functionals, which yields sharp error bounds for the quadrature remainder. The product cubature formulae are the usual tool for the approximation of a double integral over a rectangular domain. In this paper we suggest a modification of the product cubature formulae, based on blending interpolation of bivariate functions. Besides the usual point evaluations, the modified cubature formulae involve few line integrals. Our approach allows application of the Peano kernel theory for derivation of error bounds for both standard cubature formulae and their modifications. Sufficient conditions for the definiteness of the modified product cubature formulae are given, and some classes of integrands are specified, for which a product cubature formula is inferior to its modified version.