On the Bernstein Operator of S. Morigi and M. Neamtu

Abstract

Abstract. We discuss a Bernstein type operator introduced by S. Morigi and M. Neamtu for D-polynomials in the more general framework of exponential polynomials. Mathematics Subject Classification (2000). Primary 41A35; Secondary 41A50.

Optimality of generalized Bernstein operators

“…Criteria for the existence of such B n can be found in [2], [3], [20], and in the specific case of exponential polynomials, in [1] and [16]. In this section, we shall assume that B n exists (fixing the given functions), and will show that it has an extremal property analogous to that of B n .…”

“…The existence of generalized Bernstein bases raises the question of the possible existence of associated generalized Bernstein operators. This topic is studied in [1], for exponential polynomials, and in [2], [3], [16], [20], for extended Chebyshev spaces. Here we show that the operators considered in [1], [2], [3], [16], and [20], share the preceding optimality property of the classical Bernstein operator, under a suitable generalized notion of convexity.…”

Convergence of rational Bernstein operators

“…Hence p s (x) = Ax (1 − x) . By uniqueness of the representation (5) we infer that γ n−1,k (x n,k+1 − x n,k ) = Aγ n−1,k , so x n,k+1 − x n,k = A, and we arrive at the classical Bernstein operator.…”

Generalized Bernstein Operators on the Classical Polynomial Spaces

“…[11,Theorem Within the line of research followed here, previous work has focused on spaces different or more general than spaces of polynomials (cf. [15], [3], [4], [5], [13], [6], [14]), but convergence has also been studied in Müntz spaces, cf. [2], and for rational Bernstein operators, see [18].…”