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Let π
n
denote the vector space (over or
) of all polynomials of degree ≤ n. For
(where S is an appropriate set of interest), let π
n
(S) denote the class of all polynomials of degree ≤ n, all of whose zeros lie in S. Old and new problems related to the following open problem will be presented. Characterize all linear transformations (operators) T:πn(S)→ πn(S), where deg T[p] ≤ deg p for p ∈ π
n
(S).
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