Abstract
This note deals with linear operators preserving the set of positive (nonnegative) polynomials. Numerous works of prominent mathematicians in fact contain the exhaustive description of linear operators preserving the set of positive (nonnegative) polynomials. In spite of this, since this description was not formulated explicitly, it is almost lost for possible applications. In the paper we formulate and prove these classical results and give some applications. For example, we prove that there are no linear ordinary differential operators of order m ∈ ℕ with polynomial coefficients which map the set of nonnegative (positive) polynomials of degree \(\leq ( \lfloor\frac{m}{2}\rfloor+1 )\) into the set of nonnegative polynomials. This result is a generalization of a Theorem by Guterman and Shapiro.
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Katkova, O.M., Vishnyakova, A.M. (2009). Linear Operators Preserving the Set of Positive (Nonnegative) Polynomials. In: Bru, R., Romero-Vivó, S. (eds) Positive Systems. Lecture Notes in Control and Information Sciences, vol 389. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02894-6_8
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DOI: https://doi.org/10.1007/978-3-642-02894-6_8
Publisher Name: Springer, Berlin, Heidelberg
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