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
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].
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Communicated by Tim N.T. Goodman.
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Aldaz, J.M., Kounchev, O. & Render, H. Bernstein Operators for Exponential Polynomials. Constr Approx 29, 345–367 (2009). https://doi.org/10.1007/s00365-008-9010-6
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DOI: https://doi.org/10.1007/s00365-008-9010-6