On the connection of Bernstein and Kantorovich polynomials for a symmetric module function

The paper is based on the report which made by authors at the XVI International Scientific Conference “Order analysis and related problems of mathematical modeling. Operator theory and differential equations” (Vladikavkaz, September 2021). A brief review of our recent results is presented. We study the connection of Bernstein and Kantorovich polynomials for an important example with the symmetric module function. It is well known that such nonsmooth functions play a special role in approximation theory. By means of the obtained relations, the investigation of Kantorovich polynomials can be reduced to the using of the Bernstein polynomials properties. In particular, the deviation of Kantorovich polynomials from the symmetric module function is considered. In addition to accurate two-sided estimates on the interval $[0,1]$, a simple asymptotic formula for deviation is noted. The character of the convergence of Kantorovich polynomials differs from that of Bernstein polynomials give on the interval $[0,1]$. We also present new results on the convergence of Kantorovich polynomials in the complex plane. The convergence set is the same as for Bernstein polynomials. This is so-called Kantorovich compact, which limited by the lemniscate $|4z(1-z)|=1$. Everywhere here the rate of convergence of Kantorovich polynomials is established. In view of the limited size of the article, we present only the schemes of proofs. The proofs in details is planned to be given separately.