New automatic non interpolatory quadrature and cubature formulas

Given a real function $f\in C^{2k}[0,1]$, $k\ge 1$ Costabile, Gualtieri, Serra, in 1996, proposed an asymptotic expansion formula for the corresponding Bernstein polynomials $B_n(f,x)$ in terms of $h=1/n$. Previously this problem had been studied from the point of view of the search of an expansion formula in terms of the independent variable $x$. This idea was generalized to multivariate case by several authors. In the present paper we apply these results to the numerical integration problem. In this way we can build some scheme of adaptive non interpolatory quadrature for univariate and bivariate functions over rectangular and triangular domains. The theoretical results are validate by numerical tests.