Optimal recovery of a function holomorphic in a polydisc from its approximate values on a part of the skeleton

R. R. Akopyan

Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 3-29.

Résumé

We consider a series of related extremal problems for holomorphic functions in a polydisc $\mathbb{D}^m$, $m\in\mathbb{N}$. The sharp inequality $|f(z)|\le\mathscr{C}\|f\|^{\alpha_1}_{L_{\phi_1}^{p_1}(G_1)}\|f\|^{\alpha_0}_{L_{\phi_0}^{p_0}(G_0)}$, with $0{p_0}$, $p_1\le\infty$ is established between the value of a function holomorphic in $\mathbb{D}^m$ and the norms of its limit values on measurable sets $G_1$ and $G_0$, where $G_0=\mathbb{S}^m\setminus G_1$ and $\mathbb{S}^m$ is the skeleton (the Shilov boundary) of $\mathbb{D}^m$. This result is an analog of the two-constant theorem by the Nevanlinna brothers. We study conditions under which the above inequality provides us with the value of the modulus of continuity of the functional for holomorphic extension of a function on $G_1$ at a prescribed point of the polydisc. In these cases, a solution was obtained of the problem of optimal recovery of a function from approximately given values on a part of the skeleton $G_1$ and the related problem of the best approximation of the functional of the continuation of a function into a polydisk from $G_1$.

Bibliographie

[1] L. A. Aizenberg, Formuly Karlemana v kompleksnom analize. Pervye prilozheniya, Nauka, Novosibirsk, 1990 | MR

[2] R. R. Akopyan, “Analog teoremy o dvukh konstantakh i optimalnoe vosstanovlenie analiticheskikh funktsii”, Matem. sb., 210:10 (2019), 3–16 | DOI | MR | Zbl

[3] R. R. Akopyan, “Analog teoremy Adamara i svyazannye ekstremalnye zadachi na klasse analiticheskikh funktsii”, Tr. IMM UrO RAN, 26, no. 4, 2020, 32–47

[4] V. V. Arestov, “O ravnomernoi regulyarizatsii zadachi vychisleniya znachenii operatora”, Matem. zametki, 22:2 (1977), 231–244 | MR | Zbl

[5] V. V. Arestov, “Nailuchshee vosstanovlenie operatorov i rodstvennye zadachi”, Tr. Mat. in-ta AN SSSR, 189, 1989, 3–20

[6] V. V. Arestov, V. P. Gabushin, “Nailuchshee priblizhenie neogranichennykh operatorov ogranichennymi”, Izv. vuzov. Matem., 1995, no. 11, 42–68 | Zbl

[7] V. V. Arestov, “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, Uspekhi mat. nauk, 51:6 (312) (1996), 89–124 | DOI | MR | Zbl

[8] V. V. Arestov, R. R. Akopyan, “Zadacha Stechkina o nailuchshem pribli-zhenii neogranichennogo operatora ogranichennymi i rodstvennye ei zadachi”, Tr. IMM UrO RAN, 26, no. 4, 2020, 7–31

[9] N. S. Bakhvalov, “Ob optimalnosti lineinykh metodov priblizheniya operatorov na vypuklykh klassakh funktsii”, Zh. vychis. matem. i matem. fiziki, 11:4 (1971), 1014–1016 | MR

[10] V. N. Gabushin, “Nailuchshee priblizhenie funktsionalov na nekotorykh mnozhestvakh”, Matem. zametki, 8:5 (1970), 551–562 | MR

[11] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Gostekhizdat, M.–L., 1952 ; 2-е изд., Наука, М., 1966 | MR

[12] M. M. Lavrentev, “O zadache Koshi dlya uravneniya Laplasa”, Izv. AN SSSR. Ser. matem., 20:6 (1956), 819–842 | Zbl

[13] M. M. Lavrentev, V. G. Romanov, S. P. Shishatskii, Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR

[14] G. G. Magaril-Il'yaev, K. Yu. Osipenko, “Ob optimalnom vosstanovlenii funktsionalov po netochnym dannym”, Matem. zametki, 50:6 (1991), 85–93

[15] A. G. Marchuk, Optimalnye po tochnosti metody resheniya lineinykh zadach vosstanovleniya, Preprint, VTs SO AN SSSR, Novosibirsk, 1976

[16] A. G. Marchuk, K. Yu. Osipenko, “Nailuchshee priblizhenie funktsii, zadannykh v konechnom chisle tochek”, Matem. zametki, 17:3 (1975), 359–368 | Zbl

[17] U. Rudin, Teoriya funktsii v polikruge, Nauka, M., 1974 | MR

[18] S. A. Smolyak, Ob optimalnom vosstanovlenii funktsii i funktsionalov ot nikh, Dis. ... kand. fiz. mat. nauk, MGU, M., 1965

[19] S. B. Stechknn, “Nailuchshee priblizhenie lineinykh operatorov”, Matem. zametki, 1:2 (1967), 137–148

[20] S. V. Shvedenko, “Klassy Khardi i svyazannye s nimi prostranstva analiticheskikh funktsii v edinichnom kruge, polikruge i share”, Itogi nauki i tekhniki. Ser. Mat. anal., 23, VINITI, M., 1985, 3–124 | MR

[21] Ch. A. Micchelli, Th. J. Rivlin, A survey of optimal recovery Optimal estimation in approximation theory, Plenum Press, N.Y. etc., 1977, 1–54 | MR

[22] F. Nevanlinna, R. Nevanlinna, “Uber die Eigenschaften analytischer Funktionen in der Umgebung einer singularen Stelle oder Linie”, Acta Soc. Sc. Fennicae, 50, no. 5, 1922, 46 pp.

[23] K. Yu. Osipenko, Optimal Recovery of Analytic Functions, NJVA Science Publ. Inc, Huntington, 2000

[24] M. Stoll, “The space $N_*$ of holomorphie functions bounded symmetri domains”, Ann. Polon. Math., 32 (1976), 95–110 | DOI | MR | Zbl

[25] S. E. Zarantonello, “A representation of $H^p$-functions with $0 p \infty$”, Pacif. J. Math., 79:1 (1978), 271–282 | DOI | MR