Abstract

Necessary conditions are given for the Hermite–Fejér interpolation polynomials based at the zeros of orthogonal polynomials to converge in weighted Lp spaces at the Jackson rate. These conditions are known to be sufficient in the case of the generalized Jacobi polynomials.

Introduction

The first detailed study of weighted mean convergence of Hermite–Fejér interpolation based at the zeros of orthogonal polynomials was accomplished in [13] and [14], where it was shown that some of the most delicate problems associated with mean convergence of Hermite–Fejér interpolation can be approached through the general theory of orthogonal polynomials; in particular, a distinguished role is played by Christoffel functions. As opposed to Lagrange interpolation operators, Hermite–Fejér interpolation operators are not projectors, and thus in general the rate of convergence cannot be expected to equal the rate of the best approximation. Nevertheless, Jackson rates can be obtained.

Unaware of the general theory in [13] and [14] and of a variety of technical tools developed in [6], [9] and [10] (see [11] for a survey), A. K. Varma & J. Prasad in [22] investigated mean convergence of Hermite–Fejér interpolation in a particular case, namely in the case of interpolation based at the zeros of the Chebyshev polynomials. Subsequently, P. Vértesi & Y. Xu [23] wrote a paper dealing with the case of generalized Jacobi polynomials. However, their results left a significant gap between the necessary and the sufficient conditions.