On a connection of Bernstein and Kantorovich polynomials for a symmetric modulus function

Okorochkov, I.V.; Tikhonov, I.V.; Sherstyukov, V.B.

Авторы: Okorochkov I.V., Tikhonov I.V., Sherstyukov V.B.
Журнал: Siberian Mathematical Journal
Том: 64
Номер: 3
Год издания: 2023
Издательство: Maik Nauka/Interperiodica Publishing
Местоположение издательства: Russian Federation
Первая страница: 747
Последняя страница: 756
DOI: 10.1134/S0037446623030230
Аннотация: This paper bases on the authors' talk at the XVI International Scientific Conference "Order Analysis and Related Problems of Mathematical Modeling. Operator Theory and Differential Equations" (Vladikavkaz, September 2021). We briefly survey the recent results and study a connection of Bernstein and Kantorovich polynomials by the important example of the symmetric modulus function. It is well known that such nonsmooth functions play a special role in approximation theory. We reduce the study of Kantorovich polynomials to using the properties of Bernstein polynomials. In particular, we estimate the deviation of Kantorovich polynomials for the symmetric modulus function on the interval [0,1]. The rate of convergence of Kantorovich polynomials turns out different from that of Bernstein polynomials. We also present some new results on the convergence of Kantorovich polynomials in the complex plane. The convergence set in this case is the same as for Bernstein polynomials. This is the so-called Kantorovich compactum bounded by the lemniscate |4z(1-z)|=1. We establish the rate of convergence of Kantorovich polynomials everywhere on the compactum. For the sake of brevity, we give only a sketch of proofs. The full text of our study will be published separately.
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On a connection of Bernstein and Kantorovich polynomials for a symmetric modulus functionстатья