Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions

R. R. Akopyan

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 32-47

Résumé

We study several related extremal problems for analytic functions in a finitely connected domain $G$ with rectifiable Jordan boundary $\Gamma$. A sharp inequality is established between values of a function analytic in $G$ and weighted means of its boundary values on two measurable subsets $\gamma_1$ and $\gamma_0=\Gamma\setminus\gamma_1$ of the boundary: $$ |f(z_0)| \le \mathcal{C}\, \|f\|^{\alpha}_{L^{q}_{\varphi_1}(\gamma_1)}\, \|f\|^{\beta}_{L^{p}_{\varphi_0}(\gamma_0)},\quad z_0\in G, \quad 0, p\le\infty.$$ The inequality is an analog of Hadamard's three-circle theorem and the Nevanlinna brothers' two-constant theorem. In the case of a doubly connected domain $G$ and $1\le q,p\le\infty$, we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part $\gamma_1$ of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on $\gamma_1$ and the problem of the best approximation of a functional by bounded linear functionals are solved. The case of a simply connected domain $G$ has been completely investigated previously.

Détail

Keywords: analytic functions, optimal recovery of a functional, best approximation of an unbounded functional by bounded functionals, harmonic measure.

Bibliographie

[1] Akopyan R.R., “Nailuchshee priblizhenie operatora analiticheskogo prodolzheniya na klasse analiticheskikh v koltse funktsii”, Tr. Instituta matematiki i mekhaniki UrO RAN, 18:4 (2012), 3–13

[2] Akopyan R.R., “Optimalnoe vosstanovlenie analiticheskoi funktsii v dvusvyaznoi oblasti po ee priblizhenno zadannym granichnym znacheniyam”, Tr. Instituta matematiki i mekhaniki UrO RAN, 21:4 (2015), 14–19

[3] Akopyan R.R., “Analog teoremy o dvukh konstantakh i optimalnoe vosstanovlenie analiticheskikh funktsii”, Mat. sb., 210:10 (2019), 3–36 | DOI

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

[5] Arestov V.V., “Nailuchshee vosstanovlenie operatorov i rodstvennye zadachi”, cb. tr. Vsesoyuznoi shkoly po teorii funktsii (Dushanbe, avgust 1986 g.), Tr. MIAN, 189, 3–20

[6] Arestov V.V., Gabushin V.N., “Nailuchshee priblizhenie neogranichennykh operatorov ogranichennymi”, Izv. vuzov. Matematika, 1995, no. 11, 42–68

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

[8] Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A., Neravenstva dlya proizvodnykh i ikh prilozheniya, Nauk. dumka, Kiev, 2003, 591 pp.

[9] Bakhvalov N.S., “Ob optimalnosti lineinykh metodov priblizheniya operatorov na vypuklykh klassakh funktsii”, Zhurn. vychislit. matematiki i mat. fiziki, 11:4 (1971), 1014–1016

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

[11] Goluzin G.M., “Reshenie osnovnykh ploskikh zadach matematicheskoi fiziki dlya sluchaya uravneniya Laplace'a i mnogosvyaznykh oblastei, ogranichennykh okruzhnostyami (metod funktsionalnykh uravnenii)”, Mat. sb., 41:2 (1934), 246–276

[12] Goluzin G.M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, GITTL, M.; L., 1952; Наука, М., 1996, 628 с.

[13] Lavrentev M.M., Romanov V.G., Shishatskii S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980, 286 pp.

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

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

[16] Marchuk A.G., Optimalnye po tochnosti metody resheniya lineinykh zadach vosstanovleniya, preprint VTs SO AN SSSR, Novosibirsk, 1976, 29 pp.

[17] Osipenko K.Yu., “Ob optimalnykh metodakh vosstanovleniya v prostranstvakh Khardi - Soboleva”, Mat. sb., 192:2 (2001), 67–86

[18] Osipenko K.Yu., “Neravenstvo Khardi - Littlvuda - Polia dlya analiticheskikh funktsii iz prostranstv Khardi - Soboleva”, Mat. sb., 197:3 (2006), 15–34

[19] Osipenko K.Yu., “Optimalnoe vosstanovlenie lineinykh operatorov v neevklidovykh metrikakh”, Mat. sb., 205:10 (2014), 77–106

[20] Polia G., Sege G., Zadachi i teoremy iz analiza, v 2 t., v. 1, Nauka, M., 1978, 398 pp.

[21] Privalov I.I., Granichnye svoistva analiticheskikh funktsii, GITTL, M.; L., 1950, 336 pp.

[22] Smirnov V.I., “Sur les formules de Cauchy et de Green et quelques problemes qui s'y rattachent”, Izv. AN. Cer. matematicheskaya, 1932, no. 3, 337–372

[23] Smolyak S.A., Ob optimalnom vosstanovlenii funktsii i funktsionalov ot nikh, dis. ... kand. fiz.-mat. nauk, Mosk. gos. un-t, M., 1965, 152 pp.

[24] Stechkin S.B., “Neravenstva mezhdu normami proizvodnykh proizvolnoi funktsii”, Acta Sci. Math., 26:3–4 (1965), 225–230

[25] Stechkin S.B., “Nailuchshee priblizhenie lineinykh operatorov”, Mat. zametki, 1:2 (1967), 137–148

[26] Tumarkin G.Ts., Khavinson S.Ya., “O suschestvovanii v mnogosvyaznykh oblastyakh odnoznachnykh analiticheskikh funktsii s zadannym modulem granichnykh znachenii”, Izv. AN SSSR. Ser. matematicheskaya, 22:4 (1958), 543–562

[27] Tumarkin G.Ts., Khavinson S.Ya., “Klassy analiticheskikh funktsii v mnogosvyaznykh oblastyakh”, Issledovanie po sovremennym problemam teorii funktsii kompleksnogo peremennogo, cb. tr., Fizmatgiz, M., 1960, 45–77

[28] Tumarkin G.Ts., Khavinson S.Ya., “Kachestvennye svoistva reshenii ekstremalnykh zadach nekotorykh tipov”, Issledovanie po sovremennym problemam teorii funktsii kompleksnogo peremennogo, sb. tr., Fizmatgiz, M., 1960, 77–95

[29] Khavinson S.Ya., “Analiticheskie funktsii ogranichennogo vida (granichnye i ekstremalnye svoistva)”, Itogi nauki. Mat. analiz. 1963, VINITI, M., 1965, 5–80

[30] Khavinson S.Ya., “O predstavlenii ekstremalnykh funktsii v klassakh Eq cherez funktsii Grina i Neimana”, Mat. zametki, 16:5 (1974), 707–716

[31] Arestov V.V., “Best approximation of a differentiation operator on the set of smooth functions with exactly or approximately given Fourier transform”, Mathematical Optimization Theory and Operations Research (MOTOR 2019), Lecture Notes in Computer Science, 11548, eds. M. Khachay, Y. Kochetov, P. Pardalos, Springer, Cham, 2019, 434–448 | DOI

[32] Arestov V., Filatova M., “Best approximation of the differentiation operator in the space $L_2$ on the semiaxis”, J. Approx. Theory, 187:1 (2014), 65–81 | DOI

[33] Coifman R., Weiss G., “A kernel associated with certain multiply-connected domains and its applications to factorization theorems”, Studia Math., 28 (1966), 31–68

[34] Gonzalez-Vera P., Stessin M.I., “Joint spectra of Toeplitz operators and optimal recovery of analytic functions”, Constr. Approx., 36:1 (2012), 53–82 | DOI

[35] Garnett J.B., Marshall D.E., Harmonic measure, Cambridge University Press, N Y, 2005, 571 pp.

[36] DeGraw A., “Optimal recovery of holomorphic functions from inaccurate information about radial integration”, Amer. J. Comput. Math., 2:4 (2012), 258–268 | DOI

[37] Khavinson S.Ya., Kuzina T.S., “The structural formulae for extremal functions in Hardy classes on finite Riemann surfaces”, Operator Theory: Advances and Applications, 158 (2005), 37–57 | DOI

[38] Khavinson D., “Factorization theorems for different classes of analytic functions in multiply connected domains”, Pacific J. Math., 108:2 (1983), 295–318

[39] Osipenko K.Yu., Optimal recovery of analytic functions, NOVA Science Publ. Inc., Huntington, 2000, 229 pp.

[40] Osipenko K.Y., Stessin M.I., “Hadamard and Schwarz type theorems and optimal recovery in spaces of analytic functions”, Constr. Approx., 31 (2010), 37–67 | DOI

[41] Robinson R.M., “Analytic functions in circular rings”, Duke Math. J., 10:2 (1943), 341–354