Coefficients calculation of the best linear method for recovery of bounded analytic functions in a circle

Ovchintsev, Mikhail; Gusakova, E. M.

doi:10.22227/1997-0935.2014.4.44-51

Cited by 3 publications

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“…is analytical on Γ and has in K = − 1 (11) zeros. In the papers [9], [10], the following is also established (see (10), (11)…”

Section: Introductionmentioning

confidence: 99%

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Optimal recovery of derivatives of Hardy class functions

Ovchintsev

2019

E3S Web Conf.

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The paper considers the best linear method for approximating the values of derivatives of Hardy class functions in the unit circle at zero according to the information about the values of functions at a finite number of points z1,...,zn that form a regular polygon, and also the error of the best method is obtained. The introduction provides the necessary concepts and results from the papers of K.Yu. Osipenko. Some results of the studies of S.Ya. Khavinson and other authors are also mentioned here. The main section consists of two parts. In the first part of the second section, the research method is disclosed, namely, the error of the best method for approximating the derivatives at zero according to the information about the values of functions at the points z1,...,zn is calculated; the corresponding extremal function is written out. It is established that for p>1, the corresponding extremal function is unique up to a constant factor that is equal to one in modulus. For p=1, the corresponding extremal function is not unique. All such corresponding extremal functions are determined here. In the second part of the second section, it is proved that for all p (1≤p<∞), the best linear approximation method is unique, and the coefficients of the best linear recovery method are calculated. The expressions used to calculate the coefficients are greatly simplified. At the end of the paper, the obtained results are described, and possible areas for further research are indicated.

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“…is analytical on Γ and has in K = − 1 (11) zeros. In the papers [9], [10], the following is also established (see (10), (11)…”

Section: Introductionmentioning

confidence: 99%

“…Note that this paper is a continuation of [11], in which the coefficients of the best linear method for approximating the values of (0) by the values of ( 1 ), … , ( ), where ( ) ∈ ( ), are found. We need the following equality below (see [3])…”

Section: Introductionmentioning

confidence: 99%

Optimal recovery of derivatives of Hardy class functions

Ovchintsev

2019

E3S Web Conf.

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Optimal recovery of the derivate from the confined analytic function

Ovchintsev

2024

E3S Web Conf.

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In the article the author finds out the best straight-line method to recover the derivate in zero-point from the confined analytic function determined in the unit circle according to values of the function and values of its derivate in given points creating the regular polygon with zero as the center. The paper consists of three parts. The first part one contains notions and results necessary for the solution of the objective. The second part determines the measure of inaccuracy of the best approximation method and draws out the corresponding extremum function. It is established that the extremum function is the only one to an approximation of the constant multiplier modulo equal to one. The third part calculates coefficients of the best straight-line method of recovery. It is established that the best straight- linear method of recovery is the only one.

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

Ovchintsev

2019

MPCS

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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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Coefficients calculation of the best linear method for recovery of bounded analytic functions in a circle

Cited by 3 publications

References 0 publications

Optimal recovery of derivatives of Hardy class functions

Optimal recovery of derivatives of Hardy class functions

Optimal recovery of the derivate from the confined analytic function

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

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