Estimates for the moments of Bernstein polynomials

Abstract

Please cite this article in press as: J.A. Adell et al., Estimates for the moments of Bernstein polynomials, J. Math. Anal. Appl. (2015), http://dx. AbstractWe give various explicit estimates for the moments and the absolute moments of Bernstein polynomials. Asymptotically, such estimates are not far from optimality, specially for high moments. As an application, generalized Voronovskaja's formulae for Bernstein polynomials are discussed.

Sharp upper and lower bounds for the moments of Bernstein polynomials

“…[5, p. 108]). The recurrence formula given in Lemma 2 below is the key point to show Theorem 1 and was proved in [1] by using a probabilistic approach. For the sake of completeness, we include here a different proof based on the moment generating function (see [2] or [8,9])…”

New families of special numbers for computing negative order Euler numbers and related numbers and polynomials

“…The number of ways to place k rooks on a size m triangle board in three dimensions is equal to T (m + 1, m + 1 − k), where 0 ≤ k ≤ m. In [1], it is well-know that, S n (x) is a binomial random variable. That is the theory of Probability and Statistics, the binomial distribution is very useful.…”

“…(cf. [1]). For any x ∈ (0, 1), n ≥ 2, and r > 1, Adel et al [1] defined By combining (3.2) with (6.2), we arrive at the following theorem:…”

“…. Let (U k ) k≥1 be a sequence of independent distributed random variable having the uniform distribution on [0, 1] and defined by Adel et al [1]:…”

“…In [1], it is well-know that, S n (x) is a binomial random variable. That is the theory of Probability and Statistics, the binomial distribution is very useful.…”