Optimal recovery of derivatives of Hardy class functions

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The linear best method for approximating the second derivatives of Hardy class functions defined in the unit circle at zero in accordance with the information about their values in a finite number of points forming a regular polygon is found. The paper is divided into three sections. The first contains the necessary concepts and results from the work of K.Yu. Osipenko. It also recalls some results obtained by S. Ya. Havinson and other authors. In the second section, the error of the best method is calculated, and the corresponding extremal functions are written out. The third proves that the linear best approximation method is unique, and its coefficients are calculated.

Linear best method for recovering the second derivatives of Hardy class functions

Ovchintsev

2020

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Some addition to the development of mathematical support for transport navigation

Ovchintsev

2022

Formed in the 1960s, the classical methods of guaranteeing estimation are applied to provide navigation for various modes of transport. Many works of both foreign and domestic authors are devoted to the application of this method. Guaranteed estimation problems are methodologically substantiated by solving extremal problems, in particular, by the theory of optimal recovery of various classes of functions. This paper studies one of such problems, namely, the problem of optimal recovery of higher-order derivatives of functions of the Hardy class H^p,p≥1, given inthe unit circle at zero according to information about their values at the points z_1,…,z_n, forming a regular polygon centered at the origin. Thework consists of an introduction, two main sections, a presentation of the results and a conclusion. In the introduction, we present the necessary notions and results from [7]-[10]. The first section contains the error of the best method for approximating the derivatives f^((N) ) (0),1≤N≤n-1 (f(z)∈H_p ) with respect to the values f(z_1 ),….,f(z_n ). The correspondingextremal functions are also written here. In the second, the coefficients of the linear best approximation method are calculated. Examples of the optimal recovery of the first derivative of functions of the Hardy class H_H_p,p≥1 at zero from their values at a given finite number of pointsforming a regular polygon centered at the point (0,0) are given.

Optimal recovery of the derivate from the confined analytic function

Ovchintsev

2024

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.