Free interpolation in the spaces $ C^A_{r,\omega}$

N. A. Shirokov

Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 337-358

Résumé

Let the integer $r\geqslant0$ and the modulus of continuity $\omega(t)$ be fixed, and let $C^A_{r,\omega}$ be the class of all functions continuous on the closed unit disk $\overline D$, analytic on its interior $D$, and having an $\omega$-continuous $r$th derivative on $\overline D$. Consider for each $f\in C^A_{r,\omega}$ and each fixed $\zeta\in\overline D$ the polynomial in $z$ $$ P_{r,\zeta}(z;f)=\sum_{\nu=0}^r \dfrac{f^{(\nu)}(\zeta)}{\nu!} $$ (the $(r+1)$st partial sum of the Taylor series of $f$ in a neighborhood of $\zeta$). Then for any two points $\zeta_1,\zeta_2\in\overline D$ \begin{equation} \begin{gathered} |(P_{r,\zeta_1}(z)-P_{r,\zeta_2}(z))^{(\nu)}|_{z=\zeta_1}\leqslant c_f|\zeta_1-\zeta_2|^{r-\nu}\omega(|\zeta_1-\zeta_2|), \\ P_{\,\cdot\,,\,\cdot\,}(\,\cdot\,)=P_{\,\cdot\,,\,\cdot\,}(\,\cdot\,;f),\qquad 0\leqslant\nu\leqslant r. \end{gathered} \tag{1.1} \end{equation} Let $E$ be a closed subset of $\overline D$. This article contains a solution of the problem of free interpolation in $C^A_{r,\omega}$, formulated as follows: find necessary and sufficient conditions on $E$ such that for each collection $\{P_\zeta\}_{\zeta\in E}$ of $r$th-degree polynomials satisfying conditions of the type (1.1) for all $\zeta_1,\zeta_2\in E$ there is a function $f\in C^A_{r,\omega}$ with $P_\zeta(\,\cdot\,)=P_{r,\zeta}(\,\cdot\,;f)$. Bibliography: 13 titles.