APPROXIMATION OF THE DIFFERENTIATION OPERATOR ON THE CLASS OF FUNCTIONS ANALYTIC IN AN ANNULUS

Roman R. Akopyan     (Ural Federal University; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russian Federation)

Abstract


In the class of functions analytic in the annulus \(C_r:=\left\{z\in\mathbb{C}\, :\, r<|z|<1\right\}\) with bounded \(L^p\)-norms on the unit circle, we study the problem of the best approximation of the operator taking the nontangential limit boundary values of a function on the circle \(\Gamma_r\) of radius \(r\) to values of the derivative of the function on the circle \(\Gamma_\rho\) of radius \(\rho,\, r<\rho<1,\) by bounded linear operators from \(L^p(\Gamma_r)\) to \(L^p(\Gamma_ \rho)\) with norms not exceeding a number \(N\).  A solution of the problem has been obtained in the case when \(N\) belongs to the union of a sequence of intervals. The related problem of optimal recovery of the derivative of a function from boundary values of the function on \(\Gamma_\rho\) given with an error has been solved.


Keywords


Best approximation of operators, Optimal recovery, Analytic functions

Full Text:

PDF

References


  1. 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. P. 3–13. [in Russian]
  2. Akopyan R.R. Best approximation of the differentiation operator on the class of functions analytic in a strip // Proc. Steklov Inst. Math., 2015. Vol. 288, Suppl. 1. P. S5–S12. DOI: 10.1134/S0081543815020029
  3. Arestov V.V. Approximation of unbounded operators by bounded operators and related extremal problems // Russ. Math. Surv., 1996. Vol. 51, no. 6. P. 1093–1126. DOI: 10.1070/RM1996v051n06ABEH003001
  4. Stechkin S.B. Inequalities between the norms of derivatives of an arbitrary function // Acta Sci. Math., 1965. Vol. 26, no. 3–4. P. 225–230.
  5. Stechkin S.B. Best approximation of linear operators // Math. Notes, 1967. Vol. 1, no. 2. P. 91–99. DOI: 10.1007/BF01268056
  6. Osipenko K.Yu. Optimal Recovery of Analytic Functions. Huntington, NY: Nova Science, 2000.



DOI: http://dx.doi.org/10.15826/umj.2017.2.002

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.