One-sided convergence conditions for Hermite-Fejér interpolation of higher order of Lagrange type
A uniform convergence theorem is stated, generalizing a previous result for Lagrange interpolation, \textit{P. Vértesi} [Acta Sci. Math. 45, 419-428 (1983; Zbl 0549.41002)]. Given \(f\in C([-1,1])\) let \(L_{nm}^{(\alpha,\beta)}(f)\) be the Hermite-Fejér interpolation polynomial of higher order of Lagrange type. Let \(m\) be fixed, odd. If \[ \delta(f;t)=\begin{cases} o(| \ln t|^{-1}),&\text{if \(-2/m-1/2<\gamma\leq-1/2\)}\\ o(t^{m(\gamma+1/2)}), &\text{if \(-1/2<\gamma<1/m-1/2,\)} \end{cases} \] and \(|\alpha-\beta|<2/m,\) then \[ \lim_{n\to+\infty}\| L_{nm}^{(\alpha,\beta)}(f)-f\|=0. \]
Hermite-Fejér interpolation of higher order of Lagrange type
scientific article; zbMATH DE number 1267811