scholarly journals Reflections on Schur-Cohn matrices and Jury-Marden tables and classification of related unit circle zero location criteria

1996 ◽  
Vol 15 (1) ◽  
pp. 111-136 ◽  
Author(s):  
Yuval Bistritz
Keyword(s):  
Unit Circle ◽  
Zero Location

scholarly journals (1,1)-Coherent Pairs on the Unit Circle

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Garza ◽  
Francisco Marcellán ◽  
Natalia C. Pinzón-Cortés
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Linear Functional ◽  
Linear Functionals ◽  
Spectral Transformation ◽  
Coherent Pair ◽  
Coherent Pairs

A pair(𝒰,𝒱)of Hermitian regular linear functionals on the unit circle is said to be a(1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials{ϕn(x)}n≥0and{ψn(x)}n≥0satisfyϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z),an≠0,n≥1, whereϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when𝒰is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where𝒱is associated with either a positive nontrivial measure or its rational spectral transformation.

Classification of Inductive Limits of Outer Actions of ℝ on Approximate Circle Algebras

2012 ◽  
Vol 55 (1) ◽  
pp. 73-80
Author(s):  
Andrew J. Dean
Keyword(s):  
Dynamical Systems ◽  
Unit Circle ◽  
Continuous Functions ◽  
Inductive Limits ◽  
Matrix Algebras ◽  
Direct Sum

AbstractIn this paper we present a classification, up to equivariant isomorphism, of C*-dynamical systems (A, ℝ, α) arising as inductive limits of directed systems ﹛(An, ℝ, αn), φnm﹜, where each An is a finite direct sum of matrix algebras over the continuous functions on the unit circle, and the αns are outer actions generated by rotation of the spectrum.

scholarly journals Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
James McKee ◽  
Chris Smyth
Keyword(s):  
Unit Circle ◽  
Special Class ◽  
Complete Classification ◽  
Cyclotomic Polynomials ◽  
Restricted Class ◽  
Integer Polynomials ◽  
Interlacing Zeros

AbstractWe give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic Pisot polynomials. We investigate properties of this class of Pisot polynomials.

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