Geodesic
Widths of a class of analytic functions
Sbornik. Mathematics, Tome 45 (1983) no. 2, pp. 283-289 
O. G. Parfenov. Widths of a class of analytic functions. Sbornik. Mathematics, Tome 45 (1983) no. 2, pp. 283-289. http://geodesic.mathdoc.fr/item/SM_1983_45_2_a8/
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     author = {O. G. Parfenov},
     title = {Widths of a class of analytic functions},
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     volume = {45},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_45_2_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $H^2$ be the Hardy class in the unit disk, $T_r$ the circle of radius $r$ ($0) about zero, and $\alpha$ a finite Borel measure on $~T_r$. Denote by $d_n(\alpha)$ the Kolmogorov $n$th-width of the unit ball of $~H^2$ in the metric of $L_2(T_r,\alpha)$. It is proved that $$ \lim_{n\to\infty}d_n(\alpha)r^{\frac1{2}-n}=\sqrt{g(\alpha)}, $$ where $g(\alpha)$ is the geometric mean of $\alpha$ over $T_r$. The exact values of the diameters are computed for certain measures. Bibliography: 6 titles.

[1] Grenander U., Segë G., Tëplitsevy formy i ikh prilozheniya, IL, M., 1961

[2] Erokhin V. D., “O nailuchshei lineinoi approksimatsii funktsii, analiticheski prodolzhimykh s dannogo kontinuuma v dannuyu oblast”, UMN, 23:1 (1968), 91–132 | MR | Zbl

[3] Zakharyuta V. P., Skiba N. I., “Otsenki $n$-poperechkikov nekotorykh klassov funktsii, analiticheskikh na rimanovykh poverkhnostyakh”, Matem. zametki, 19:6 (1976), 899–911 | MR | Zbl

[4] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[5] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[6] Parfenov O. G., Poperechniki nekotorykh klassov analiticheskikh funktsii i asimptoticheskoe povedenie spektra integralnykh operatorov s gladkimi yadrami, Dis. na soiskanie uch. st. kand. fiz.-matem. nauk, izd-vo LGU, Leningrad, 1979 | MR