Chebotarev's theorem on the stability of a function

Chebotarev's theorem on the stability of a function is a generalization of the Hermite - Biehler theorem to the case of entire functions. Named after the Soviet mathematician Nikolai Chebotaryov .

Wording

Whole function $f$ ${\ displaystyle f}$ if and only if it is very stable when the corresponding functions $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ constitute a real pair and at least at one point of the real axis the function $gh'-g'h$ ${\ displaystyle gh'-g'h}$ positive.

Explanation

Here the whole function is considered the function $f(z)$ ${\ displaystyle f (z)}$ integrated variable $z$ ${\ displaystyle z}$ decomposing in a power series: $f(z)=a_{0}+a_{1}z+\ldots +a_{n}z^{n}+\ldots$ ${\ displaystyle f (z) = a_ {0} + a_ {1} z + \ ldots + a_ {n} z ^ {n} + \ ldots}$ converging for all values $z$ ${\ displaystyle z}$ . An entire function is stable if it has no roots with a positive real part. Functions $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ are defined as follows. Substituting in $f(z)$ ${\ displaystyle f (z)}$ instead $z$ ${\ displaystyle z}$ pure imaginary number $i\omega$ ${\ displaystyle i \ omega}$ we get a complex number $f(i\omega )=g(\omega )+ih(\omega )$ ${\ displaystyle f (i \ omega) = g (\ omega) + ih (\ omega)}$ . Whole functions $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ constitute a real pair if for any real $\lambda$ ${\ displaystyle \ lambda}$ and $\mu$ ${\ displaystyle \ mu}$ all function roots $\lambda g+\mu h$ ${\ displaystyle \ lambda g + \ mu h}$ real. If function $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ make up a real pair, then the roots of these functions are interleaved. Polynomial roots $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ with real coefficients are interspersed if both polynomials have only real and simple roots and between any two adjacent roots of one polynomial one and only one root of the other polynomial is contained.

Literature

Postnikov M.M. Stable Polynomials - M .: Nauka, 1981. - P. 129.

Chebotarev's theorem on the stability of a function

Wording

Explanation

Literature

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