A discussion of simultaneous approximation of derivatives by lagrange interpolation

Abstract

We investigate here the simultaneous approximation of a function f e Cq [−1,1] by a polynomial Pnf interpolating f and of f(1) ,… f(q) by the respective derivatives of Pnf. The nodes for the interpolation consist of n “basic” nodes in (−1,1) for n = 1,2,… augmented by “added” nodes for each n, converging to −1 at the rate and other “added” nodes converging to 1 at the same rate. We show for instance that, if q is even , in which Ln is the interpolation on the “basic” nodes. Similar results hold if q is odd. No assumptions need be made about the nodes other than what is already stated. If, for example, one uses for each n the zero set of the Chebychev polynomial cos(n arc cos x) as the basic nodes, then , and our results thus obtain the rate of convergence which is “best possible.” Similarly, the prescribed rate of convergence to +1 and −1 for the added nodes is the most minimal assumption which gives good approximation of derivatives. In particular, the added nodes can lie partially or completely on top o...