Quantitative Bounds for Hurwitz Stable Polynomials under Multiplier Transformations
نویسنده
چکیده
We extend results of Borcea and Brändén to quantitatively bound the movement of the zeros of Hurwitz stable polynomials under linear multiplier operators. For a multiplier operator T on polynomials of degree up to n, we show that the ratio between the maximal real part of the zeros of a Hurwitz stable polynomial f and that of its transform T [f ] is minimized for the polynomial (z + 1). For T operating on polynomials of all degrees, we use the classical PólyaSchur theorem to derive an asymptotic formula for the maximal real part of the zeros of T [(z+1)], and consider the action of T on classes of entire functions. We develop interlacing controls on the transformations of real-rooted polynomials. On the basis of numerical evidence, we conjecture several further quantitative controls on the zeros of Hurwitz stable polynomials under multiplier operators.
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