Quasi-interpolations with interpolation property

Interpolation and quasi-interpolation are very important methods for function approximation. But both of them have their respective disadvantages. The interpolation function can fit the given sample points, but some oscillation may arise as in the case of the Lagrange interpolation. The quasi-interpolation function, for example, the Bernstein quasi-interpolation function, satisfies the convergence condition, but does not keep fitting the given sample points. In this paper, we present a method to construct a quasi-interpolation operator with certain interpolation property.