On the Structure of Kergin Interpolation for Points in General Position

For n + 1 points in IR, in general position, the Kergin polynomial interpolant of C functions may be extended to an interpolant of C functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k, 0 ≤ k ≤ d − 1. We show that the polynomials dual to the functionals depending on derivatives of order k are multi-ridge functions of d− k variables and moreover, that the polynomials dual to the purely interpolating functionals (k = 0) are always harmonic.