Approximate Hermite quasi-interpolation
  In this paper we derive approximate quasi-interpolants when the values of a function $u$ and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants which provide high order approximations for solutions to elliptic differential equations with constant coefficients. 
Flavia Lanzara, Vladimir Maz'ya, Gunther Schmidt