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
Abstract. We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0 , and a second function f 1 such that f 1 /f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C[a, b] → U n with strictly increasing nodes, fixing f 0 , f 1 ∈ U n . If U n ⊂ U n+1 and U n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n+1 : C[a, b] → U n+1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 , f 1 , ..., f n is a basis of U n such that the linear span of f 0 , .., f k is an extended Chebyshev space over [a, b] for each k = 0, ..., n, then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold
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