# R.R. Akopyan. Optimal recovery of a function analytic in a half-plane from approximately given values on a part of the straight-line boundary... P. 19-33

Full text (in Russian)

Let $\mathcal{H}^p(\Pi_+,\phi)$ be the class of functions analytic in the upper half-plane $\Pi_+$ and belonging to the universal Hardy class $N_*$ with boundary values from $L^p_\phi(\mathbb{R})$ with a weight $\phi$, and let $Q^p(\Pi_+,\mathbb{I},\phi)$ be the class of function $f\in \mathcal{H}^p(\Pi_+,\phi)$ such that $\|f\|_{L^p_\phi(\mathbb{R}\setminus\mathbb{I})}\le 1$, where $\mathbb{I}$ is a finite open interval or a half-line from $\mathbb{R}$ and $1\le p\le\infty.$ On the class $Q^p(\Pi_+,\mathbb{I},\phi)$, we consider the problem of optimal recovery of the value of a function at a point $z_0\in\Pi_+$ from its approximately given limit boundary values on $\mathbb{I}$ in the norm $L^p_\phi(\mathbb{I})$ and the related problem of the best approximation of a functional by linear bounded functionals. Explicit solutions of these problems are written: an extremal function, optimal recovery method, and best approximation functional. On the class $Q^p(\Pi_+,\mathbb{R}_+,\psi)$, $\psi(z)=1/|z|$, we solve the problem of optimal recovery of a function on a ray $\gamma=\{z\,:\,\arg z=\varphi_0\}$ with respect to the norm $L^p_\psi(\gamma)$ from its approximately given limit boundary values on~$\mathbb{R}_+$ in the norm $L^p_\psi(\mathbb{R}_+)$ and the related problem of the best approximation of an operator by linear bounded operators. For $f\in\mathcal{H}^p(\Pi_+,\psi)$, we obtain the exact inequality $$\|f\|_{L^p_{\psi}(\gamma)}\le \|f\|_{L^{p}_{\psi}(-\infty, 0)}^{{\varphi_0}/{\pi}}\, \|f\|_{L_{\psi}^{p}(0, +\infty)}^{1-{\varphi_0}/{\pi}}.$$

Keywords: optimal recovery of an operator, best approximation of an unbounded operator by bounded operators, analytic function

Revised November 14, 2018

Accepted November 19, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Roman Razmikovich Akopyan, Ural Federal University, Yekaterinburg, 620000 Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: RRAkopyan@mephi.ru

REFERENCES

1.   Ajzenberg L.A. Carleman formulas in complex analysis. First applications. Novosibirsk: Nauka, 1990, 248 p. (in Russian).

2.   Akopyan R.R, Best approximation of the operator of analytic continuation on the class of functions analytic in a strip. Trudy Inst. Mat. i Mekh. UrO RAN, 2011, vol. 17, no. 3, pp. 46—54 (in Russian).

3.   Akopyan R.R. Optimal recovery of analytic functions from boundary conditions specified with error. Math. Notes, 2016, vol. 99, no. 2, pp. 177–182. doi: 10.1134/S000143461601020X .

4.   Akopyan R.R. Best approximation of the functional of the analytic continuation from a part of the boundary. Proc. 18th Int. Saratov Winter School “Contemporary Problems of Function Theory and Their Applications”. Saratov: Nauchnaya kniga Publ., 2016, pp. 25–26 (in Russian). ISBN: 978-5-9758-1623-8 .

5.   Akopyan R.R. Optimal recovery of a derivative of an analytic function from values of the function given with an error on a part of the boundary. Analysis Math., 2018, vol. 44, no. 1, pp. 3–19. doi: 10.1007/s10476-018-0102-7 .

6.   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 .

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

8.   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 .

9.   Arestov V., Filatova M. Best approximation of the differentiation operator in the space $L_2$ on the semiaxis. J. Approx. Theory, 2014, vol. 187, no. 1, pp. 65–81. doi: 10.1016/j.jat.2014.08.001 .

10.   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 .

11.   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 .

12.   Lavrent’ev M.M., Romanov V.G., Shishatskii S.P. Nekorrektnye zadachi matematicheskoj fiziki i analiza (III-posed problems of mathematical physics and analysis). Providence: American Math. Soc., 1986, Ser. Transl. Math. Monographs, vol. 64, 290 p.

13.   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 .

14.   Micchelli Ch.A., Rivlin Th.J. A survey of optimal recovery. In: Optimal estimation in approximation theory. N.Y. etc.: Plenum Press, 1977, pp. 1–54. doi: 10.1007/978-1-4684-2388-4_1 .

15.   Magaril-Il’yaev G.G., Tikhomirov V.M., Osipenko K.Yu. Indefinite knowledge about an object and accuracy of its recovery methods. Probl. Inf. Transm., 2003, vol. 39, no. 1, pp. 104–118. doi: 10.1023/A:1023686600253 .

16.   Osipenko K.Yu. Optimal Recovery of Analytic Functions. Huntington: NOVA Science Publ.Inc., 2000, 229 p. ISBN: 1-56072-821-3 .

17.   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 .

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.   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 .