Generalized Strang-Fix condition for scattered data quasi-interpolation

Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang-Fix condition, which is related to non-stationary quasi-interpolation. Based on the discussion of the generalized Strang-Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too.