We extend Clenshaw-Curtis quadrature to the square in a nontensorial way, by using Sloan's hyperinterpolation theory and two families of points recently studied in the framework of bivariate (hyper)interpolation, namely the Morrow-Patterson-Xu points and the Padua points. The construction is an application of a general approach to product-type cubature, where we prove also a relevant stability theorem. The resulting cubature formulas turn out to be competitive on nonentire integrands with tensorproduct Clenshaw-Curtis and Gauss-Legendre formulas, and even with the few known minimal formulas.2000 AMS subject classification: 65D32.

We give an asymptotic expansion in powers of n −1 of the remainder ∞ j=n f j z j , when the sequence f n has a similar expansion. Contrary to previous results, explicit formulas for the computation of the coefficients are presented. In the case of numerical series (z = 1), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of S(z; j 0 , ν, a, b, p) = ∞ j=j 0 z j (j + b) ν−1 (j + a) −p , which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series.

We propose an adaptive algorithm which extends Chebyshev series approximation to bivariate functions, on domains which are smooth transformations of a square. The method is tested on functions with different degrees of regularity and on domains with various geometries. We show also an application to the fast evaluation of linear and nonlinear bivariate integral transforms.

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