Bernstein Operators for Exponential Polynomials

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 {|Im λ _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

$$B_{n}f:=\sum_{k=0}^{n}\alpha _{k}f(t_{k})p_{n,k}(x)$$

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∈C[a,b].