On Chebyshev-type integral quasi-interpolation operators

Spline quasi-interpolants on the real line are approximating splines to given functions with optimal approximation orders. They are called integral quasi-interpolants if the coefficients in the spline series are linear combinations of weighted mean values of the function to be approximated. This paper is devoted to the construction of new integral quasi-interpolants with compactly supported piecewise polynomial weights. The basic idea consists of minimizing an expression appearing in an estimate for the quasi-interpolation error. It depends on how well the quasi-interpolation operator approximates the first non-reproduced monomial. Explicit solutions as well as some numerical tests in the B-spline case are given.