Pointwise radial minimization: Hermite interpolation on arbitrary domains

In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We generalize this and minimize over derivatives of polynomials of arbitrary odd degree. We analyze the cubic case, which assumes first derivative boundary data and show that the minimization has a unique, infinitely smooth solution with cubic precision. We have not been able to prove that the solution satisfies the Hermite interpolation conditions but numerical examples strongly indicate that it does for a wide variety of planar domains and that it behaves nicely.