A necessary condition for convergence of interpolating parabolic and cubic splines

N. L. Zmatrakov

Matematičeskie zametki, Tome 19 (1976) no. 2, pp. 165-178

Résumé

Let the sequence of nets $\Delta_n$ be such that $\lim\limits_{n\to\infty}\max\limits_ih_i^{(n)}=0$, where $h_i^{(n)}$ are the lengths of the segments of a net. The bound $\max\limits_{|i-j|=1}\frac{h_i^{(n)}}{h_j^{(n)}1-\alpha}\le R<\infty$ is necessary in order that interpolating parabolic and cubic splines converge for any function in $C(\alpha=0)$ or $C_\alpha(0<\alpha<1)$, where $C_\alpha$ is the class of functions satisfying a Lipschitz condition of order $\alpha$. It is also shown that this bound cannot essentially be weakened.