Divergence of interpolation processes on sets of the second category

Al. A. Privalov

Matematičeskie zametki, Tome 18 (1975) no. 2, pp. 179-183.

Résumé

$C([0,1])$ is the space of real continuous functions $f(x)$ on $[0,1]$ and $\omega(\delta)$ is a majorant of the modulus of continuity $\omega(f,\delta)$, satisfying the condition $\varlimsup\limits_{n\to\infty}\omega(1/n)\ln n=\infty$. A solution is given to a problem of S. B. Stechkin: for any matrix $\mathfrak M$ of interpolation points there exists an $f(x)\in C([0,1])$, $\omega(f,\delta)=o\{\omega(\delta)\}$ whose Lagrange interpolation process diverges on a set $\mathscr E$ of second category on $[0,1]$.