Schur’s Theorem for Hurwitz Polynomials

In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented.

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Zeros of the Wigner distribution and the short-time Fourier transform

Revista Matemática Complutense ◽

10.1007/s13163-019-00335-w ◽

2019 ◽

Vol 33 (3) ◽

pp. 723-744 ◽

Cited By ~ 1

Author(s):

Karlheinz Gröchenig ◽

Philippe Jaming ◽

Eugenia Malinnikova

Keyword(s):

Fourier Transform ◽

Bessel Functions ◽

Wigner Distribution ◽

Short Time Fourier Transform ◽

Step Functions ◽

Wigner Distributions ◽

Hurwitz Polynomials ◽

Totally Positive ◽

Short Time ◽

Positive Functions

AbstractWe study the question under which conditions the zero set of a (cross-) Wigner distribution W(f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson’s theorem for the positivity of the Wigner distribution and to Hardy’s uncertainty principle. We then construct a class of step functions S so that the Wigner distribution $$W(f,\mathbf {1}_{(0,1)})$$ W ( f , 1 ( 0 , 1 ) ) always possesses a zero $$f\in S \cap L^p$$ f ∈ S ∩ L p when $$p<\infty $$ p < ∞ , but may be zero-free for $$f\in S \cap L^\infty $$ f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

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On a theorem of Schur

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201006457 ◽

2001 ◽

Vol 28 (8) ◽

pp. 455-460 ◽

Cited By ~ 5

Author(s):

Peter Hilton

Keyword(s):

Nilpotent Group ◽

Commutator Subgroup ◽

Schur’S Theorem

We study the ramifications of Schur's theorem that, ifGis a group such thatG/ZGis finite, thenG′is finite, if we restrict attention to nilpotent group. HereZGis the center ofG, andG′is the commutator subgroup. We use localization methods and obtain relativized versions of the main theorems.

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