Error bounds for Gaussian quadrature rules using linear kernels

It is well-known that the remaining term of a n-point Gaussian quadrature depends on the 2n-order derivative of the integrand function. Discounting the fact that calculating a 2n-order derivative requires a lot of differentiation, the main problem is that an error bound for a n-point Gaussian quadrature is only relevant for a function that is 2n times differentiable, a rather stringent condition. In this paper, by defining some specific linear kernels, we resolve this problem and obtain new error bounds (involving only the first derivative of the weighted integrand function) for all Gaussian weighted quadrature rules whose nodes and weights are pre-assigned over a finite interval. The advantage of using linear kernels is that their L-1-norm, L-infinity-norm, maximum and minimum can easily be computed. Three illustrative examples are given in this direction.