HERMITE-FEJER TYPE INTERPOLATION AND KOROVKINS THEOREM

In this note we consider Hermite-Fejer interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejer type operators F(mr)(a+r,beta) and special values of alpha,beta it was proved by the first author in recent papers that one has uniform convergence on the whole interval [-1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejer interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejer type interpolation polynomials F(mr)(a+r,beta) f, f-epsilon C([-1,1]), converge pointwise to f for arbitrary alpha,beta > -1. The convergence is uniform on [-1 + delta,1 - delta].