GENERALIZ GENERALIZATION OF THE H TION OF THE HARDY CLASS FOR AN ASS FOR ANALYTIC FUNCTIONS

Introduction. Quoting from a well-known American mathematician Lipman Bers [1] “It would be tempting to rewrite history and to claim that quasiconformal transformations have been discovered in connection with gas-dynamical problems. As a matter of fact, however, the concept of quasiconformality was arrived at by Grotzsch [2] and Ahlfors [3] from the point of view of function theory”. The present work is devoted to the theory of analytic solutions of the Beltrami equation which directly related to the quasi-conformal mappings. The function is, in general, assumed to be measurable with almost everywhere in the domain under consideration. Solutions of equation (1) are often referred to as analytic functions in the literature. Research methods.

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

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.

Optimal recovery of derivatives of Hardy class functions

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.

Optimal Recovery of Functions of the Class Hp,p (1≤ p ≤ ∞) in the Unit Circle

In the below work the problem of optimal recovery of functions in Hardy class is covered. Namely, by the values ​​of these functions in a finite number of points lying in the unit circle determined their value at a given point. Coefficients of the linear best approximation method and error of the best method are calculated. The functions are considered with some given weight function.