Best methods of approximation and interpolation of differentiable functions in~the space $C_{[-1,1]}$

V. M. Tikhomirov

Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 275-289

Résumé

In this paper are computed the $n$-diameter of the class $$ W_r=\{f(x):|f^{(r-1)}(x)-f^{(r-1)}(x')|\leqslant|x-x'|,|x|,|x'|\leqslant1\} $$ of functions defined on $[-1,1]$ in $C_{[-1,1]}$. This problem reduces to the variational problem \begin{gather*} \lambda_{nr}=\inf||x||,\\ x^{(r+1)}=2\sum_{k=1}^m(-1)^{k+1}\delta(t-t_k),\qquad-1\leqslant t_1\leqslant\dots\leqslant t_m\leqslant1,\quad m\leqslant n,\\ x^r(t)\equiv-1,\qquad t\leqslant-1, \end{gather*} whose solution is described in Theorem 1 of the paper. Bibliography: 6 titles.