scholarly journals Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula

2020 ◽  
Author(s):  
Jacob S Christiansen ◽  
Benjamin Eichinger ◽  
Tom VandenBoom
Keyword(s):  
Unit Circle ◽  
Functional Model ◽  
Rational Functions ◽  
Almost Periodic ◽  
Absolutely Continuous ◽  
Magic Formula ◽  
Carathéodory Functions ◽  
Möbius Transforms ◽  
Rotational Phase ◽  
Isospectral Torus

Abstract We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum ${\textsf{E}}$ and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as ${\textsf{E}}$-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them “MCMV.” Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.

The Zero Distribution of Orthogonal Rational Functions on the Unit Circle

1997 ◽  
Vol 49 (4) ◽  
pp. 810-819
Author(s):  
K. Pan
Keyword(s):  
Unit Circle ◽  
Potential Theory ◽  
Rational Functions ◽  
Orthogonal Rational Functions ◽  
Zero Distribution

AbstractRational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied. We use the potential theory to discuss the zeros distribution for the orthogonal rational functions.

scholarly journals Salem Numbers and Pisot Numbers via Interlacing

2012 ◽  
Vol 64 (2) ◽  
pp. 345-367 ◽  
Author(s):  
James McKee ◽  
Chris Smyth
Keyword(s):  
Unit Circle ◽  
Rational Functions ◽  
General Construction ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Limit Points ◽  
Zeros And Poles

Abstract We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the “obvious” limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

scholarly journals SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

2007 ◽  
Vol 50 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unit Circle ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
The Other ◽  
Complex Numbers ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
The One

AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

scholarly journals Quadrature formulas on the unit circle based on rational functions

1994 ◽  
Vol 50 (1-3) ◽  
pp. 159-170 ◽  
Author(s):  
Adhemar Bultheel ◽  
Erik Hendriksen ◽  
Pablo González-Vera ◽  
Olav Njåstad
Keyword(s):  
Unit Circle ◽  
Rational Functions ◽  
Quadrature Formulas

Some Properties of Rational Functions with Prescribed Poles

1999 ◽  
Vol 42 (4) ◽  
pp. 417-426 ◽  
Author(s):  
Abdul Aziz-Ul-Auzeem ◽  
B. A. Zarger
Keyword(s):  
Unit Circle ◽  
Rational Function ◽  
Rational Functions ◽  
Sharp Result

AbstractLet P(z) be a polynomial of degree not exceeding n and let where |aj| > 1, j = 1, 2,…,n. If the rational function r(z) = P(z)/W(z) does not vanish in |z| < k, then for k = 1 it is known thatwhere B(Z) = W*(z)/W(z) and . In the paper we consider the case when k > 1 and obtain a sharp result. We also show thatwhere , and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

scholarly journals ORTHOGONAL RATIONAL FUNCTIONS AND INTERPOLATORY PRODUCT RULES ON THE UNIT CIRCLE

Analysis ◽  
2000 ◽  
Vol 20 (2) ◽  
pp. 99-120 ◽  
Author(s):  
A. Bultheel ◽  
P. González-Vera ◽  
E. Hendriksen ◽  
O. Njåstad
Keyword(s):  
Unit Circle ◽  
Rational Functions ◽  
Orthogonal Rational Functions ◽  
Product Rules
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