Lagrange Interpolation on a Lattice: Bounding Derivatives by Divided Differences

Fix an integer n > 0. For a multivariate function defined on a (not necessarily rectangular) lattice, an extension is constructed to have, ∀k ≤ n, derivatives of total degree k that are bounded by the function’s tensor product divided differences of total degree k times a constant independent of the lattice and the function. The extension is locally constructed, can have any prescribed smoothness, and reproduces polynomials of degree < n in each variable.