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

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