Generalized Popoviciu expansions for Bernstein polynomials of a rational module

I. V. Tikhonov ; V. B. Sherstyukov ; D. G. Tsvetkovich

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 71-117.

Résumé

We prove that Bernstein polynomials for simple nonsmooth functions such as a rational module can be represented as special sums of a regular structure called “generalized Popoviciu decompositions.” To write generalized expansions, a certain formalism based on combinatorial calculations is developed. Based on the formulas obtained, we obtain a complete description of the convergence set of Bernstein polynomials of a rational module. The connection between the Popoviciu expansions and the distribution of zeros of Bernstein polynomials on the complex plane is discussed. In conclusion, a number of additional, new relations for Bernstein polynomials of a rational module are presented.

Détail

Keywords: Bernstein polynomial, piecewise linear function, generalized Popoviciu expansion, Kantorovich lemniscate
Mots-clés : rational module, domain of convergence, distribution of zeros of a polynomial.

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