Simultaneous approximation by Lagrange-Hermite Interpolation

What is it about?

In a paper Y. E. Muneer (Acta Math. Hung. 49 (1987), 293-305) has obtained convergence results for the simultaneous approximation of the first i derivatives of a sufficiently smooth function defined on [-1,1] by the derivatives of its Lagrange interpolant on Cebysev nodes. To overcome tha bad convergence near the endpoints he considered additional Hermite interpolation at the endpoints ±1, which gives a great improvement in the order of approximation. In a joint work of K. Balázs with the first of the authors (J. Approx. Theory 60 (1990), 231-244) the results of Muneer were generalized to the case of arbitrary node systems, the order of convergence beeing estimated in terms of the norms of the interpolation operators corresponding to the given system of nodes. It is our aim here to investigate the Hermite interpolant H_mf of a function f in C^q[-1,1], where the derivatives at ±1 are interpolated up to an order r-1. We give here error estimates in the maximum norm over [-1,1] for the i-th derivative of f - H_mf in all possible cases 0 ≤ i,r-1 ≤ q. As a special choice of nodes we investigate the roots of Jacobi polynomials.

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