Generalized Popoviciu Expansions for Bernstein Polynomials of a Rational Module

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 263; no. 4; pp. 532 - 578
• Main Authors: Tikhonov, I. V., Sherstyukov, V. B., Tsvetkovich, D. G.
• Format: Journal Article
• Language: English
• Published: New York Springer US 09.05.2022; Springer; Springer Nature B.V
• Subjects: Combinatorial analysis; Functions (mathematics); Historical structures; Mathematical analysis; Mathematics; Mathematics and Statistics; Modules; Polynomials; 30E10; 41A25; piecewise linear function; generalized Popoviciu expansion; 41A10; domain of convergence; distribution of zeros of a polynomial; rational module; Kantorovich lemniscate; Bernstein polynomial; 30C15
• ISSN: 1072-3374; 1573-8795; 1573-8795
• DOI: 10.1007/s10958-022-05948-z

We study Bernstein polynomials for simple nonsmooth rational module functions. We show that these polynomials can be represented as special sums of regular structure. For historical reasons, the representations found are naturally called “generalized Popoviciu expansions.” To write generalized expansions, we develop a special formalism based on combinatorial calculations. Based on the formulas obtained, we propose a complete description of the character of convergence of the Bernstein polynomials studied in the complex plane. We also discuss the relationship of generalized Popoviciu expansions with the distribution of zeros of Bernstein polynomials. In the final part of the paper, we present additional new relations for Bernstein polynomials of the rational module function.