Abstract
Error estimates for cubature rules are usually given in terms of partial derivatives (Peano-Sard) or in terms of analyticity properties (Davis-Hammerlin). The approximation method has found little attention. This method can be refined by using biorthogonal systems (BOGS). We present these BOGS estimates both for cubature rules which are exact for Pm (the space of bivariate polynomials of total degree notgreater than m) and for cubature rules which are exact for Pk,1 (the space of bivariate polynomials with degree in x notgreater than k and degree in y notgreater than 1). As BOGS there will be used bivariate Chebyshev polynomials and the Fourier coefficient functionals. Furthermore, for product rules we get another error bound by reducing the cubature error to the quadrature errors (Nikolskii) and taking the BOGS estimates for quadrature rules. As application we treat Clenshaw-Curtis product rules.
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