Some results in the theory of interpolation using the Legendre polynomial and its derivative (Q1089539)

Let \(\xi_ 1,...,\xi_ n\) be the zeros of the Legendre polynomial \(P_ n\) of degree n, and \(x_ 1,...,x_{n-1}\) the zeros of \(P_ n'\). The author deals with the following quasi-Hermite-Fejér interpolation process: Given \(f\in C[-1,1]\), find a polynomial \(H_ n(f)\) of degree \(4n+1\) such that \(H_ n(f,\xi_ k)=f(\xi_ k)\), \(H_ n'(f,\xi_ k)=0\), \(k=1,...,n\), \(H_ n(f,x_ k)=f(x_ k)\), \(H_ n'(f,x_ k)=0\), \(k=1,...,n-1\), \(H_ n(f,\pm 1)=f(\pm 1)\), \(H_ n'(f,\pm 1)=0\). Such a polynomial is constructed, and a pointwise error estimate for this interpolation process is given. The author also considers quasi-Hermite- Fejér interpolation of another type.