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