Bernstein polynomials are studied on a symmetric interval. Basic relations connected with Bernstein polynomials for a standard module function are received. By the Templ's formula we establish recurrence relations from which the Popoviciu's expansion is derived. Suitable formulas for the first and second derivatives are found. As a result an explicit algebraic form for Bernstein polynomials is obtained. We also notice some corollaries.
Ставится вопрос о явной алгебраической записи полиномов Бернштейна по степеням независимой переменной. Кратко обсуждается общая постановка задачи на произвольном отрезке a,b. Для полноты картины напоминаются формулы Вигерта, действующие для коэффициентов полиномов Бернштейна на стандартном отрезке 0,1. Вцентре внимания сейчас другой случай симметричного отрезка 1,1, что представляет несомненный интерес для теории аппроксимации. Вработе найдены выражения, регулирующие образование коэффициентов полиномов Бернштейна на 1,1. Для интерпретации ответа потребовалось ввести новые числовые объекты специальные трапеции Паскаля. Они строятся аналогично классическому треугольнику по своим начальным и краевым условиям. Страпециями Паскаля связаны разнообразные соотношения, во многом обобщающие привычные комбинаторные тождества. Вработе проведено систематическое исследование подобных свойств составлена сводка основных формул. Полученные результаты находят применение при изучении поведения коэффициентов полиномов Бернштейна на 1,1. Так, например, оказывается, что есть универсальная связь двух коэффициентов a2m,m(f) иam,m(f), действующая при всех minmathbbN для любой функции fin C1,1. Витоге установлено существенное отличие картины на 1,1 от случая стандартного отрезка 0,1. Намечен ряд перспективных тем для дальнейших исследований, часть из которых активно проводится впоследнее время.
The subject of the paper is closely related to one general direction in the approximation theory, within which the growth rate of the coefficients of algebraic polynomials is studied for uniform approximations of continuous functions. The classical Bernstein polynomials play an important role here. We study in detail a model example of Bernstein polynomials for the standard modulus function on a symmetric interval. The question under consideration is the growth rate of of the coefficients in these polynomials with an explicit algebraic representation. It turns out that in the first fifteen polynomials the growth of the coefficients is almost not observed. For the next polynomials the situation changes, and coefficients of exponential growth appear. Our main attention is focused on the behaviour of the maximal coefficient, for which exact exponential asymptotics and corresponding two-sided estimates are established (see Theorem 2). As it follows from the obtained result, the maximal coefficient has growth 2 /2 / 2 , where is the index of the Bernstein polynomial. It is shown that the coefficients equidistant from the maximal one have a similar growth rate (for details, see Theorem 3). The group of the largest coefficients is located in the central part of the Bernstein polynomials but at the ends the coefficients are sufficiently small. The behavior of the sum of absolute values of all coefficients is also considered. This sum admits an explicit expression that is not computable in the sense of traditional combinatorial identities. On the base of a preliminary recurrence relation, we succeeded to obtain the exact asymptotics for the sum of absolute values of all coefficients and to give the corresponding two-sided estimates (see Theorem 4). The growth rate of the sum is 2 /2 / 3/2. In the end of the paper, we compare this result with a general Roulier estimate and new related problems are formulated.
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