A superconvergent cubic spline quasi-interpolant and application (Q898028)

It is well known that a spline quasi-interpolant is an approximation operator obtained as a liner combination of basic spline functions with bounded support. The coefficients of this quasi-interpolant are computed so that it is exact on a space of polynomials of a given degree. A new technique to get a superconvergence phenomenon of cubic spline quasi-interpolants at the knots of an uniform partition is presented. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. A new simple quadrature rule based on integrating a cubic spline quasi-interpolant with superconvergence is studied. Error estimates for smooth functions are given, too. Numerical results are given to illustrate the theoretical ones.