A natural formulation of quasi-interpolation by multivariate splines

A quasi-interpolation formnula based on discrete function values is givein in the form of a multivariate spline series that yields the local approximation order characterized by the Fix-Strang conditions. This formula can be considered as a partial sum of the formal Neumann series expansion of the formal interpolation operator, and hence, justifies that "quasi-interpolation" is indeed an appropriate terminology for such an approximation formula.