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Generalizations of the Hermite-Biehler theorem. (English) Zbl 0949.15037

The authors generalize the Hermite-Biehler theorem which characterizes the Hurwitz stability of a real polynomial \(\delta(s).\)
Let \(\delta(s)\) be a real polynomial of degree \(n\) and let \(l\) and \(r\) denote the numbers of roots in the left half and the right half of the complex plane, respectively. The signature \(l-r\) of \(\delta(s)\) determines \(l\) and \(r\) since \(l+r=n.\) Let \(i\) be the square root of \(-1\) and let \(\omega\) be a real number. Then \(\delta(i\omega)=p(\omega)+iq(\omega).\) The authors find a formula expressing the signature of \(\delta(s)\) involving \(p\) and \(q\) and zeros of these functions.
Reviewer: E.Ellers (Toronto)

MSC:

15A63 Quadratic and bilinear forms, inner products
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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