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On the properties of Sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature

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Abstract

Each Sard kernels theorem supplies multiple error estimates for a bounded linear functional, provided the underlying multivariate function is sufficiently smooth. The error estimation due to Sard generally concerns the \(L_1\)-norms of Sard kernels and the supremum norms of different partial derivatives of a given function. This article presents a verified method for computing Sard error constants. To assist the verified computation, we examine relevant properties of Sard kernels for general bounded linear functionals of bivariate functions. The derived \(L_1\)-norms, together with interval Taylor arithmetic, make the multiple error estimates possible. We demonstrate the flexibility and superiority of Sard kernels method by different numerical examples that concern non-product cubature for bivariate functions.

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Acknowledgments

The support of this work by the National Science Council of Taiwan under Project No. NSC 97-2115-M-415-002- and NSC 100-2115-M-415-002- is greatly acknowledged.

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  1. Department of Applied Mathematics, National Chiayi University, No 300, University Rd., Chiayi, 600, Taiwan

    Chin-Yun Chen

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  1. Chin-Yun Chen
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Chen, CY. On the properties of Sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature. Numer. Math. 122, 603–643 (2012). https://doi.org/10.1007/s00211-012-0471-y

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