Geodesic
Closed ideals in algebras of functions analytic in the disk and smooth up to its boundary
Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 425-445 
F. A. Shamoyan. Closed ideals in algebras of functions analytic in the disk and smooth up to its boundary. Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 425-445. http://geodesic.mathdoc.fr/item/SM_1994_79_2_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A complete description is obtained for closed ideals in algebras of holomorphic functions $f$ on the unit disk such that $$ \bigl|f^{(n)}(\zeta_1)-f^{(n)}(\zeta_2)\bigr|=o\bigl(\omega(|\zeta_1-\zeta_2|)\bigr) \qquad (|\zeta_1-\zeta_2|\to 0). $$ Here $n$ is a nonnegative integer, and $\omega$ an arbitrary nonnegative nondecreasing subadditive function on $(0,+\infty)$.

[1] Carleman T., Les fonctions quasianalytiques, Paris, 1926

[2] Shilov G. E., “Ob odnom granichnom svoistve analiticheskikh funktsii”, Uchenye zapiski MGU, Matematika, 3 (1949), 126–128

[3] Rudin W., “The closed ideals in an algebra of analytic functions”, Canad. J. Math., 9 (1957), 426–434 | MR | Zbl

[4] Korenblyum B. I., “Zamknutye idealy algebry $A^n$”, Funkts. analiz i ego prilozh., 6:3 (1972), 38–53 | MR

[5] Shamoyan F. A., “Zamknutye idealy v algebrakh analiticheskikh funktsii, gladkikh vplot do granitsy”, Izv. AN Arm. SSR, Matematika, 15:4 (1980), 323–331 | MR | Zbl

[6] Shamoyan F. A., “Zamknutye idealy algebry $\lambda_\omega ^{(n)}$”, Zapiski nauchnykh seminarov LOMI AN SSSR, 126, Nauka, L., 1983, 203–205 | MR

[7] Khruschev S. V., “Vsyakii ideal koltsa $C^n_A$ yavlyaetsya glavnym”, Zapiski nauchn. seminarov LOMI AN SSSR, 65, Nauka, L., 1976, 149–160 | MR | Zbl

[8] Pavlov B. S., “O nesamosopryazhennom operatore Shredingera”, Problemy mat. fiziki, no. 3, LGU, 1968

[9] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, Gostekhizdat, M., 1950

[10] Smirnov V. I., Lebedev N. A., Konstruktivnaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1964 | MR

[11] Taylor B. A., Williams D. L., “Ideals in rings of analytic functions with smooth boundary values”, Canad. J. Math., 22:6 (1970), 1266–1283 | MR | Zbl

[12] Tamrazov P. M., “Konturnye i telesnye strukturnye svoistva golomorfnykh funktsii kompleksnogo peremennogo”, UMN, XXVIII:1(169) (1973), 131–161 | MR

[13] Taylor B. A., Williams D. L., “Zeros of lipschitz functions analytic in the unit disc”, Mich. Math. J., 18:22 (1971), 24–27 | MR

[14] Korenblyum B. I., “O funktsiyakh, golomorfnykh v kruge, gladkikh vplot do ego granitsy”, DAN SSSR, 200:1 (1971), 24–27 | MR | Zbl

[15] Beurling A., “On two problems concerning linear transformations in Hilbert space”, Acta Math., 81 (1949), 239–255 | DOI | MR | Zbl