On the properties of Sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature
Sard kernel theorems concern the result of applying a bounded linear functional to a Taylor expansion of a bivariate, sufficiently smooth function defined on a rectangle. \textit{A. H. Stroud} [Approximate calculation of multiple integrals. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. (1971; Zbl 0379.65013)] has used Sard kernels to estimate the errors of certain tensor product and non-product cubature rules. In this paper, the author extends these results and presents a verified method for computing Sard error constants. The flexibility and superiority of the Sard kernel method for certain tensor product and non-product cubature rules are shown. First the author discusses some basic properties of Sard kernels that are useful for the verified computation. Then relevant properties of Sard kernels for cubature rules are examined. The ideas for computing Sard error constants by interval arithmetic are described and corresponding algorithms are presented. The error constants for a non-product cubature rule are calculated by the proposed algorithms. Numerical examples are given.
scientific article; zbMATH DE number 6117691
scientific article