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.

Bibliographie

[1] Vinogradov S. A., Khavin V. P., “Svobodnaya interpolyatsiya v $H^\infty$ i nekotorykh drugikh klassakh funktsii. I; II”, Zap. nauchn. sem. LOMI, 47 (1974), 15–54 ; 56 (1976), 12–58 | MR | Zbl | MR | Zbl

[2] Dynkin E. M., “Svobodnaya interpolyatsiya v klassakh Gëldera”, DAN SSSR, 236:4 (1977), 785–788 | MR

[3] Dynkin E. M., “Mnozhestva svobodnoi interpolyatsii dlya klassov Gëldera”, Matem. sb., 109(151) (1979), 107–128 | MR

[4] Bruna J., “Les eusembles d'interpolation des $A^p(D)$”, C. r. Acad. scient., AB 290:1 (1980), A25–27 | MR

[5] Kotochigov A. M., “Interpolyatsiya analiticheskimi funktsiyami, gladkimi vplot do granitsy”, Zap. nauchn. sem. LOMI, 30 (1972), 167–169 | Zbl

[6] Kotochigov A. M., Svobodnaya interpolyatsiya v prostranstvakh analiticheskikh funktsii, gladkikh vplot do granitsy, Dis. na soiskanie uch. st. kand. fiz.-matem. nauk, Leningrad, 1981 | Zbl

[7] Alfors L., Lektsii po kvazikonformnym otobrazheniyam, Mir, M., 1969 | MR

[8] Belinskii P. P., Obschie svoistva kvazikonformnykh otobrazhenii, Novosibirsk, 1974 | MR

[9] Dzyadyk V. K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[10] Shirokov N. A., “Standartnye idealy algebry $H_n^1$”, Funkts. analiz, 13:1 (1979), 86–87 | MR | Zbl

[11] Carleson L., “An interpolation problem for bounded analytic functions”, Amer. J. Math., 80:4 (1958), 921–930 | DOI | MR | Zbl

[12] Vasyunin V. I., “Bezuslovno skhodyaschiesya spektralnye razlozheniya i zadachi interpolyatsii”, Tr. matem. in-ta im. V. A. Steklova, SKhKhKh (1977), 5–49

[13] Shirokov N. A., “Idealy i faktorizatsiya v algebrakh analiticheskikh funktsii, gladkikh vplot do granitsy”, Tr. matem. in-ta im. V. A. Steklova, SKhKhKh (1977), 196–222