Hurwitz rational functions

S., Barkovsky, Yury; Tyaglov, Mikhail

doi:10.1016/j.laa.2011.03.062

Cited by 6 publications

(10 citation statements)

References 6 publications

(17 reference statements)

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“…We are starting from the following short but useful observation. 4 Lemma 1. Let φ be a function holomorphic in C + := {z ∈ C : Im z > 0} with values in C + and let ψ be a fixed holomorphic branch of…”

Section: Connection Between ℛ-Functions and Univalent Functionsmentioning

confidence: 99%

“…This, in turn, implies that the function Re ψ(x) in (b), (c) and (d) of Lemma 5 and (b) of Lemma 6 may lose the extremum and become monotonic. In cases (a) and (c) of Lemma 6, Re ψ(I) is only a semi-infinite interval of the real line, instead of the equality Im ψ(I) = R. 4 Location of α-points in the closed upper half-plane Lemma 8. Let functions φ 1 (z), φ 2 (z) be of the form (7)…”

Section: ∂ ∂Rmentioning

confidence: 99%

“…XV], [22,4,6]). With a proper extension of the notion of stability, this criterion extends to entire (see [6]), rational (see [4]) and further towards meromorphic functions. Furthermore, if H(z) is a polynomial and we additionally allow the ratio f (z) g(z) to have positive zeros and poles, then we will obtain the "generalized Hurwitz" polynomials as introduced in [22].…”

Section: Introductionmentioning

confidence: 99%

“…[13, Ch. XV], [22,4,6]). With a proper extension of the notion of stability, this criterion extends to entire (see [6]), rational (see [4]) and further towards meromorphic functions.…”

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confidence: 99%

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One helpful property of functions generating Pólya frequency sequences

Dyachenko

2016

Comput. Methods Funct. Theory

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In this work we study the solutions of the equation z p R(z k ) = α with nonzero complex α, integer p, k and R(z) generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of z p R(z k ) − α are simple (or at most double in the case Im α k = 0) and split evenly among the sectors { j k π Arg z j+1 k π}, j = 0, . . . , 2k − 1. Our approach rests on the fact that z(ln z p/k R(z)) ′ is an ℛ-function (i.e. maps the upper half of the complex plane into itself).This result guarantees the same localization to zeros of entire functions f (z k ) + z p g(z k ) and g(z k ) + z p f (z k ) provided that f (z) and g(−z) have genus 0 and only negative zeros. As an application, we deduce that functions of the form ∑ ∞ n=0 (±i) n(n−1)/2 a n z n have simple zeros distinct in absolute value under a certain condition on the coefficients a n 0. This includes the "disturbed exponential" function corresponding to a n = q n(n−1)/2 /n! when 0 < q 1, as well as the partial theta function corresponding to a n = q n(n−1)/2 when 0 < q q * ≈ 0.7457224107.

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Section: Connection Between ℛ-Functions and Univalent Functionsmentioning

confidence: 99%

Section: ∂ ∂Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[13, Ch. XV], [22,4,6]). With a proper extension of the notion of stability, this criterion extends to entire (see [6]), rational (see [4]) and further towards meromorphic functions.…”

mentioning

confidence: 99%

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One helpful property of functions generating Pólya frequency sequences

Dyachenko

2016

Comput. Methods Funct. Theory

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“…Our future goals include using this work for generalizations of these classical results to univariate entire and meromorphic functions, including questions on Pólya frequency sequences, Hurwitz rational functions, and entire functions with all real zeros [98,12]. One particular area of interest is the class of so-called generalized Hurwitz polynomials, which is a useful large class containing all Hurwitz stable polynomials.…”

Section: Introductionmentioning

confidence: 99%

Structured Matrices, Continued Fractions, and Root Localization of Polynomials

Holtz¹,

Tyaglov²

2012

SIAM Rev.

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We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde, and other structured matrices, Stietjes-and Jacobitype continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.Introduction. This paper offers a glimpse at several classical topics going back to Descartes, Gauss, Stieltjes, Hermite, Hurwitz, and Sylvester (see, e.g., [19,80,28,81,82]), all connected by the idea that behavior of polynomials can be analyzed via algebraic constructs involving their coefficients. Thus a number of interrelated algebraic constructs were built, including Hurwitz, Toeplitz, and Hankel matrices, the corresponding quadratic forms, and the corresponding continued fractions. The linear-algebraic properties of these objects were shown to be intimately related to root localization of polynomials such as stability, whereby the zeros of a polynomial avoid a specific half-plane, or hyperbolicity, whereby the zeros lie on a specific line. These methods gave rise to several well-known tests of root localization, such as the Routh-Hurwitz algorithm or the Lienard-Chipart test.In the 20th century, this line of research was developed further by Schur [24], Pólya [78], Krein [59,60,34], and others (see, e.g., [43]), leading to important notions of total positivity, Pólya frequency sequences, stability preservers, etc. A very important part of that research effort was devoted to entire functions, in particular, the

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Stability analysis

Bavafa-Toosi¹

2019

Introduction to Linear Control Systems

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Hurwitz rational functions

Cited by 6 publications

References 6 publications

One helpful property of functions generating Pólya frequency sequences

One helpful property of functions generating Pólya frequency sequences

Structured Matrices, Continued Fractions, and Root Localization of Polynomials

Stability analysis

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