O.V. Akopyan, R.R. Akopyan. Optimal recovery on classes of functions analytic in a annulus ... P. 7-23

Let $C_{r,R}$ be a annulus with boundary circles $\gamma_r$ and $\gamma_R$ centered at zero; its inner and outer radii are $r$ and $R$, respectively. On the class of functions analytic in the annulus $C_{r,R}$ with finite $L^2$-norms of the angular limits on the circle $\gamma_r$ and of the $n$th derivatives (of the functions themselves for $n=0$) on the circle $\gamma_R$, we study interconnected extremal problems for the operator $\psi_{\rho}^m$ that takes the boundary values of a function on $\gamma_r$ to its restriction (for $m=0$) or the restriction of its $m$th derivative (for $m>0$) to an intermediate circle $\gamma_\rho$, $r<\rho<R$. The problem of the best approximation of $\psi_{\rho}^m$ by linear bounded operators from $L^2(\gamma_r)$ to $C(\gamma_\rho)$ is solved. A method for the optimal recovery of the $m$th derivative on a intermediate circle $\gamma_\rho$ from $L^2$-approximately given values of the function on the boundary circle $\gamma_r$ is proposed and its error is found. The Hadamard—Kolmogorov exact inequality, which estimates the uniform norm of the $m$th derivative on an intermediate circle $\gamma_\rho$ in terms of the $L^2$-norms of the limit boundary values of the function and the $n$th derivative on the circles $\gamma_r$ and $\gamma_R$, is derived.

Keywords: analytic functions, Hadamard three-circle theorem, Kolmogorov's inequality, optimal recovery

Received February 10, 2023

Revised February 27, 2023

Accepted February 27, 2023

Olga Vladimirovna Akopyan, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: olga_akopjan@rambler.ru

Roman Razmikovich Akopyan, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: RRAkopyan@mephi.ru


1.   Arestov V.V. Uniform regularization of the problem of calculating the values of an operator. Math. Notes, 1977, vol. 22, no. 2, pp. 618–626. doi: 10.1007/BF01780971

2.   Arestov V.V. Optimal recovery of operators and related problems. Proc. Steklov Inst. Math., 1990, no. 4, pp. 1–20.

3.   Arestov V.V., Gabushin V.N. Best approximation of unbounded operators by bounded ones. Russian Math. (Iz. VUZ), 1995, vol. 39, no. 11, pp. 38–63.

4.   Arestov V.V. Approximation of unbounded operators by bounded operators and related extremal problems. Russian Math. Surveys, 1996, vol. 51, no. 6, pp. 1093–1126. doi: 10.1070/RM1996v051n06ABEH003001

5.   Babenko V.F., Korneichuk N.P., Kofanov V.A. and Pichugov S.A. Neravenstva dlya proizvodnykh i ikh prilozheniya [Inequalities for derivatives and their applications]. Kiev: Naukova Dumka Publ., 2003, 590 p. ISBN: 966-00-0074-4 .

6.   Magaril-Il’yaev G.G., Osipenko K.Yu. Optimal recovery of functionals based on inaccurate data. Math. Notes, 1991, vol. 50, no. 6, pp. 1274–1279. doi: 10.1007/BF01158269

7.   Osipenko K.Yu. Optimal recovery of analytic functions. Huntington: NOVA Science Publ. Inc., 2000, 229 p. ISBN: 1-56072-821-3 .

8.   Osipenko K.Yu. Optimal recovery of linear operators in non-Euclidean metrics. Sb. Math., 2014, vol. 205, no. 10, pp. 1442–1472. doi: 10.1070/SM2014v205n10ABEH004425

9.   Pólya G., Szegö G. Problems and theorems in analysis. Vol. 1. Berlin, Springer, 1972, 392 p. doi: 10.1007/978-1-4757-1640-5

10.   Akopyan R.R. Best approximation for the analytic continuation operator on the class of analytic functions in a ring. Trudy Inst. Mat. i Mekh. UrO RAN, 2012, vol. 18, no. 4, pp. 3–13 (in Russian).

11.   Robinson R.M. Analytic functions in circular rings. Duke Math. J., 1943, vol. 10, no. 2, pp. 341–354. doi: 10.1215/S0012-7094-43-01031-2

12.   Goluzin G.M. Geometric theory of functions of a complex variable. Ser. Transl. Math. Monogr., vol. 26, Providence, R.I.: American Math. Soc., 1969, 676 p. ISBN: 978-0-8218-1576-2 . Original Russian text published in Goluzin G.M. Geometricheskaya teoriya funktsii kompleksnogo peremennogo: uchebnoe posobie, Moscow, Leningrad: Nauka GITTL Publ., 1952, 628 p.

13.   Akopyan R.R. Approximation of the differentiation operator on the class of functions analytic in an annulus. Ural Math. J., 2017, vol. 3, no. 2, pp. 6–13. doi: 10.15826/umj.2017.2.002

14.   Tumarkin G.Ts., Khavinson S.Ya. Qualitative properties of solving extreme problems of certain types. In: Issled. Sovrem. Probl. Teor. Funkcij Kompleks. Peremen. Moscow, Fizmatgiz Publ., 1960, pp. 77–95 (in Russian).

15.   Khavinson S.Ya. Analytic functions of bounded type. Itogi Nauki. Mat. Anal., 1963, VINITI, Moscow, 1965, pp. 5–80 (in Russian).

16.   Khavinson S.Ya. Representation of extremal functions in the classes Eq in terms of Green’s and Neumann’s functions. Math. Notes, 1974, vol. 16, no. 5, pp. 1018–1023. doi: 10.1007/BF01149790

17.   Khavinson S.Ya., Kuzina T.S. The Structural formulae for extremal functions in Hardy classes on finite Riemann surfaces. Operator Theory: Advances and Applications, 2005, vol. 158, pp. 37–57. doi: 10.1007/3-7643-7340-7_4

18.   Osipenko K.Y., Stessin M.I. Hadamard and Schwarz type theorems and optimal recovery in spaces of analytic functions. Constr. Approx., 2010, vol. 31, pp. 37–67. doi: 10.1007/s00365-009-9043-5

19.   Akopyan R.R. An analogue of the two-constants theorem and optimal recovery of analytic functions. Sb. Math., 2019, vol. 210, no. 10, pp. 1348–1360. doi: 10.1070/SM8952

20.   Osipenko K.Yu. On optimal recovery methods in Hardy–Sobolev spaces. Sb. Math., 2001, vol. 192, no. 2, pp. 225–244. doi: 10.1070/SM2001v192n02ABEH000543

21.   Gonzalez-Vera P., Stessin M.I. Joint spectra of Toeplitz operators and optimal recovery of analytic functions. Constr. Approx., 2012, vol. 36, no. 1, pp. 53–82. doi: 10.1007/s00365-012-9169-8

22.   DeGraw A. Optimal recovery of holomorphic functions from inaccurate information about radial integration. Amer. J. Comput. Math., 2012, vol. 2, no. 4, pp. 258–268. doi: 10.4236/ajcm.2012.24035

23.   Osipenko K.Yu. The Hardy–Littlewood–Polya inequality for analytic functions in Hardy–Sobolev spaces. Sb. Math., 2006, vol. 197, no. 3, pp. 315–334. doi: 10.1070/SM2006v197n03ABEH003760

24.   Ovchintsev M. Linear best method for recovering the second derivatives of Hardy class functions. In: E3S Web of Conferences, 2020, vol. 164, article no. 02013. doi: 10.1051/e3sconf/202016402013

25.   Taikov  L.V. Kolmogorov-type inequalities and the best formulas for numerical differentiation, Math. Notes, 1968, vol. 4, no. 2, pp. 631–634. doi: 10.1007/BF01094964

26.   Akopyan R.R. Best approximation of differentiation operators on the Sobolev class of functions analytic in a strip. Sib. Elektron. Mat. Izv., 2021, vol. 18, no. 2, pp. 1286–1298 (in Russian). doi:10.33048/semi.2021.18.098

27.   Babenko V., Babenko Yu., Kriachko N., Skorokhodov D. On Hardy–Littlewood–Pólya and Taikov type inequalities for multiple operators in Hilbert spaces. Anal. Math., 2021, vol. 47, pp. 709–745. doi: 10.1007/s10476-021-0104-8

28.   Vvedenskaya E.V., Osipenko K.Yu. Discrete analogs of Taikov’s inequality and recovery of sequences given with an error. Math. Notes, 2012, vol. 92, no. 4, pp. 473–484. doi: 10.1134/S0001434612090192

29.   Shadrin A.Yu. Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes $W_2^m$. Math. Notes, 1990, vol. 48, no. 4, pp. 1058–1063. doi: 10.1007/BF01139609

Cite this article as: O.V. Akopyan, R.R. Akopyan. Optimal recovery on classes of functions analytic in a annulus, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 7–23; Proceedings of the Steklov Institute of Mathematics  (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S4–S19.