Piecewise Hermite interpolation via barycentric coordinates: In memory of Prof. Carlo Ciliberto

Piecewise interpolation methods, as spline or Hermite cubic interpolation methods, define the interpolating function by means of polynomial pieces and ensure that some regularity conditions hold at the break-points. In this paper, starting from a previous work, where we proposed a class of piecewise interpolating functions whose expression depends on the barycentric coordinates, we extend that approach to obtain an interpolant for which Hermite constraints are assigned. The underlying idea is to define the interpolant on each subinterval, by weighting two Taylor polynomials, centered at the endpoints, of a function that generated the Hermite conditions. The weights depend on a suitable function of the barycentric coordinates of the evaluation point. We show that the interpolating function inherits the properties of regularity from such a weight function. Moreover, we study the interpolation error and provide bounds in terms of both the degree of the Taylor polynomials and properties of the weight function. Numerical experiments confirm the theoretical results and show the reliability of the bounds provided for the interpolation error. © 2015, Università degli Studi di Napoli "Federico II".