On the simultaneous approximation of derivatives by Lagrange and Hermite interpolation

Abstract

We investigate here, for a positive integer q, simultaneous approximation of the first q derivatives of a function by the derivatives of its Lagrange interpolant, and then we augment this procedure by Hermite interpolation at the endpoints of the interval, obtaining a great improvement in the quality of approximation. In both cases, we estimate the quality of simultaneous approximation in terms of the norm of an associated Lagrange interpolation, and the estimates are thus valid for any sequence of interpolations by polynomials of successively higher degree. This communication continues work begun by K. Balázs and generalizes a recent work of Muneer Yousif Elnour, who treats simultaneous approximation with nodes at the zeroes of the Tchebycheff polynomials. Our efforts to obtain results which are independent of the choice of nodes have also led to some interesting consequences of a theorem of Gopengauz on simultaneous approximation.