A lacunary interpolation algorithm on arbitrary points

Here, the following lacunary interpolation problem is considered: to find the polynomial which together with its second and third derivatives agrees on arbitrary points with the corresponding values of a given function. The representation of the polynomial depends on the solution of a linear algebraic system. The method is constructed on this observation. The error analysis shows that it behaves like the corresponding Lagrange interpolation problem with an equivalent number of conditions. Some applications are shown.