On generalized Hermite-Fejer interpolation of Lagrange type on the Chebyshev nodes

For f is an element of C [-1, 1], let H-m,H-n(f, x) denote the (0, 1,..., m) Hermite-Fejer (HF) interpolation polynomial off based on the Chebyshev nodes. That is, H-m,H-n(f, x) is the polynomial of least degree which interpolates f(x) and has it:; first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind, in this paper a precise pointwise estimate for the approximation error \H-2m,H-n(f, x) -f(x)\ is developed, and an equiconvergence result for Lagrange and (0, 1,..., 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0,1,...,2m) HF interpolation methods on the Chebyshev nodes, is convergent for all f is an element of C[ -1, 1]. (C) 2000 Academic Press.