Algorithms and convergence for Hermite interpolation based on extended Chebyshev nodal systems

Abstract

The Chebyshev nodal systems play an important role in the theory of Hermite interpolation on the interval - 1 , 1 ] . For the cases of nodal points corresponding to the Chebyshev polynomials of the second kind U n ( x ) , the third kind V n ( x ) and the fourth kind W n ( x ) , it is usual to consider the extended systems, that is, to add the endpoints -1 and 1 to the nodal system related to U n ( x ) , to add -1 to the nodal system related to V n ( x ) and to add 1 to the nodal system related to W n ( x ) . The interpolation methods that are usually used in connection with these extended nodal systems are quasi-Hermite interpolation and extended Hermite interpolation, and it is well known that the performance of these two great methods is quite good when it comes to continuous functions.This work attempts to complete the theory concerning these extended Chebyshev nodal systems. For this, we have obtained a new formulation for the Hermite interpolation polynomials based upon barycentric formulas. The feature of this approach is that the derivatives of the function at the endpoints of the interval are also prescribed. Further, some convergence results are obtained for these extended interpolants when apply to continuous functions.