Peano kernel behaviour and error bounds for symmetric quadrature formulas

For symmetric quadrature formulas, sharper error bounds are generated by a formulation of the Peano Kernel theory which fully exploits the symmetry properties. Formulas which use nodes outside the integration interval and/or derivative data are taken into consideration. We prove also that, for any symmetric formula, the Peano Kernels of a sufficiently large degree do not change sign in a suitable interval. This result allows a finite expansion of the truncation error for any regular integrand function.