Bernstein Operators for Exponential Polynomials

Abstract

Abstract. Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ 0 , ..., λ n . Assume that the set U n of all solutions of the equation Lf = 0 is closed under complex conjugation. If the length of the interval [a, b] is smaller than π/M n , where M n := max {|Imλ j | : j = 0, ..., n}, then there exists a basis p n,k , k = 0, ...n, of the space U n with the property that each p n,k has a zero of order k at a and a zero of order n − k at b, and each p n,k is positive on the open interval (a, b) . Under the additional assumption that λ 0 and λ 1 are real and distinct, our first main result states that there exist points a = t 0 < t 1 < ... < t n = b and positive numbers α 0 , .., α n , such that the operator

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

“…see [1]. Therefore there exists under this assumption a Bernstein basis in E (λ0,...,λn) for {a, b}.…”

“…In this paper we want to discuss and compare a recent result of S. Morigi and M. Neamtu in [13] about the construction and convergence of a Bernstein operator for so-called D-polynomials with our recent results in [1] for exponential polynomials. Let us recall that the space of exponential polynomials for given complex numbers λ 0 , ..., λ n is defined by…”

“…In [1] we have discussed the existence of Bernstein bases for exponential spaces E (λ0,...,λn) with complex exponents λ 0 , ..., λ n . We say that E (λ0,...,λn) is closed under complex conjugation if f ∈ E (λ0,...,λn) implies that the complex conjugate f is in E (λ0,...,λn) .…”

“…In particular, if λ 0 , ..., λ n are real then M n is just zero and one obtains the well known result that E (λ0,...,λn) is an extended Chebyshev system over any interval [a, b] . In passing, we note that Bernstein bases for exponential spaces can be defined in a simple and recursive way, for details see [1]. In this paper we shall consider the case of equidistant exponents, i.e.…”

“…By Proposition 4 in [1], the existence of a Bernstein basis consisting of real-valued functions is equivalent to the property that E (λ0,...,λn) is closed under complex conjugation. In [13] S. Morigi and M. Neamtu introduced a Bernstein operator based on D-polynomials.…”