scholarly journals Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

1999 ◽  
Vol 97 (2) ◽  
pp. 384-395
Author(s):  
Konstantin Yu Osipenko
Keyword(s):  
Analytic Functions ◽  
Optimal Recovery ◽  
Hardy Classes

scholarly journals About the Optimal Recovery of Derivatives of Analytic Functions from Their Values at Points That Form a Regular Polygon

2019 ◽  
pp. 30-38
Author(s):  
Mikhail Ovchintsev
Keyword(s):  
Approximation Method ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Best Approximation ◽  
Extremal Function ◽  
Regular Polygon ◽  
Optimal Recovery ◽  
Bounded Analytic Functions ◽  
The Best Approximation ◽  
Derivatives Of

In this paper, the author solves the problem of optimal recovery of derivatives of bounded analytic functions defined at the zero of the unit circle. Recovery is performed based on information about the values of these functions at points z1, ... , zn , that form a regular polygon. The article consists of an introduction and two sections. The introduction talks about the necessary concepts and results from the works of Osipenko K.Yu. and Khavinson S.Ya., that form the basis for the solution of the problem. In the first section, the author proves some properties of the Blaschke product with zeros at the points z1, ... , zn. After this, the error of the best approximation method of the derivatives f(N)(0), 1 ≤ N ≤ n − 1, by the values f(z1), ... , f(zn) is calculated. In the same section he gives the corresponding extremal function. In the second section, the uniqueness of the linear best approximation method is established, and then its coefficients are calculated. At the end of the article, the formulas found for calculating of the coefficients are substantially simplified.

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