Bernstein Operators for Exponential Polynomials

Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues lambda(0),...,lambda(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 pi/M-n, where M-n := max{vertical bar Im lambda(j)vertical bar : 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 lambda(0) and lambda(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 alpha(0),..., alpha(n), such that the operator B(n)f := (n)Sigma(k=0) alpha(k)f(t(k)) p(n,k)(x) satisfies B(n)e(lambda)j(x) = e(lambda)j(x), for j = 0, 1. The second main result gives a sufficient condition guaranteeing the uniform convergence of B(n)f to f for each f is an element of C[a, b].