On growth rate of coefficients in Bernstein polynomials for the standard modulus function on a symmetric interval

Abstract

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 $2^{n/2}/\,n^2$, where $n$ 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 $2^{n/2}/\,n^{3/2}$. In the end of the paper, we compare this result with a general Roulier estimate and new related problems are formulated.