About the optimal recovery of derivatives of analytic functions from their values at points that form a regular polygon

M. P. Ovchintsev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 22 (2019) no. 4, pp. 30-38

Résumé

In this paper, the author solves the problem of optimal recovery of derivatives of bounded analytic functions defined at zero of the unit circle. Recovery is performed based on information about the values of these functions at points $z_1, \dots,z_n$, that form a regular polygon. The article consists of an introduction and two sections. The introduction discusses the necessary concepts and results from the works of K.Yu. Osipenko and S.Ya. Khavinson, 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 points $z_1, \dots,z_n$. After this, the error of the best approximation method of derivatives $f^{(N)}(0)$, $1\leq N\leq n-1$, by values $f(z_1), \dots,f(z_n)$ is calculated. In the same section the author 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 the coefficients are substantially simplified.

Détail

Keywords: optimal recovery, best approximation method, error of the best method, extremal function, linear best method, coefficients of the linear best method.