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This article is cited in 1 scientific paper (total in 1 paper)
Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations
M. A. Petrosovaa, I. V. Tikhonovb, V. B. Sherstyukovc a Moscow State Pedagogical University, 14 Krasnoprudnaya Str., Moscow 107140, Russia
b Lomonosov Moscow State University, 1 Leninskie gory, Moscow 119991, Russia
c National Research Nuclear University MEPhI, 31 Kashirskoe shosse, Moscow 115409, Russia
Abstract:
We pose the question of explicit algebraic representation for Bernstein polynomials. The general statement of the problem on an arbitrary interval $[a,b]$ is briefly discussed. For completeness, we recall Wigert formulas for the polynomials coefficients on the standard interval $[0,1]$. However, the focus of the paper is the case of the symmetric interval $[-1,1]$, which is of fundamental interest for approximation theory. The exact expressions for the coefficients of Bernstein polynomials on $[-1,1]$ are found. For the interpretation of the results we introduce a number of new numerical objects named Pascal trapeziums. They are constructed by analogy with a classical triangle, but with their own “initial” and “boundary” conditions. The elements of Pascal trapeziums satisfy various relations which remind customary combinatorial identities. A systematic research on such properties is fulfilled, and summaries of formulas are given. The obtained results are applicable for the study of the behavior of the coefficients in Bernstein polynomials on $[-1,1]$. For example, it appears that there exists a universal connection between two coefficients $a_{2m,m}(f)$ and $a_{m,m}(f)$, and this is true for all $m\in\mathbb N$ and for all functions $f\in C[-1,1]$. Thus, it is set up that the case of symmetric interval $[-1,1]$ is essentially different from the standard case of $[0,1]$. Perspective topics for future research are proposed. A number of this topics is already being studied.
Key words:
Bernstein polynomials, symmetric interval, Pascal trapeziums, combinatorial relations.
Received: 21.06.2016 Revised: 22.06.2019
Citation:
M. A. Petrosova, I. V. Tikhonov, V. B. Sherstyukov, “Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations”, Vladikavkaz. Mat. Zh., 21:3 (2019), 68–86
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https://www.mathnet.ru/eng/vmj700 https://www.mathnet.ru/eng/vmj/v21/i3/p68
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Abstract page: | 274 | Full-text PDF : | 90 | References: | 32 |
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