Bernstein operators for exponential polynomials

Resumo

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 Bnf:= ∑nk=0αkf(tkP n,k(x) satisfies Bneλjx, 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]. © 2008 Springer Science+Business Media, LLC.