Certain unbounded Hermite-Fejér interpolatory polynomial operators
The paper deals with the generalized Hermite-Fejér interpolatory polynomial \(L_{m,n}[f,x]\) given by \(L_{m,n}[f,x_ k]=f(x_ k)\), \(L^{(j)}_{m,n}[f,x_ k]=0\), \(k=1,2,\dots,n\), \(j=1,2,\dots,m-1\), \(f\in C[-1,1]\), where the \(x_ k\), \(k=1,2,\dots,n\) denote the zeroes of the \(n\)-th Chebyshev polynomial \(T_ n\), \(n=1,2,3,\dots\). The result of the paper is to prove that the operator \(L_{m,n}[f]\) is unbounded for all odd numbers \(m\). The case of even numbers \(m\) was already considered in the author's, Hermite-Fejér interpolation prescribing higher order derivatives, in: Progress in Approximation Theory (1991).
scientific article; zbMATH DE number 149655