On stability ofm-variateC 1 interpolation

A simplicial mesh (triangulation) is constructed that generalizes the two-dimensional 4-direction mesh to ℝ ^m . This mesh, with symmetric, shift-invariant values at the vertices, is shown to admit a bounded C¹ interpolant if and only if the alternating sum of the values at the vertices of any 1-cube is zero. This implies that interpolation at the vertices of an m-dimensional, simplicial mesh by a C¹ piecewise polynomial of degree m+1 with one piece per simplex is unstable.