Asymptotic Behavior of Random Vandermonde Matrices With Entries on the Unit Circle

2009 ◽  
Vol 55 (7) ◽  
pp. 3115-3147 ◽  
Author(s):  
Øyvind Ryan ◽  
MÉrouane Debbah
Keyword(s):  
Asymptotic Behavior ◽  
Unit Circle ◽  
Vandermonde Matrices

scholarly journals On the Asymptotic Behavior of Functions Harmonic in a Disc

1966 ◽  
Vol 28 ◽  
pp. 187-191
Author(s):  
J. E. Mcmillan
Keyword(s):  
Asymptotic Behavior ◽  
Unit Circle ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Continuous Curve ◽  
Open Unit Disc ◽  
Boundary Path

Let D be the open unit disc, and let C be the unit circle in the complex plane. Let f be a (real-valued) function that is harmonic in D. A simple continuous curve β: z(t) (0≦t<1) contained in D such that |z(t)|→1 as t→1 is a boundary path with end (the bar denotes closure).

scholarly journals On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

2020 ◽  
Author(s):  
Stefan Kunis ◽  
Dominik Nagel
Keyword(s):  
Unit Circle ◽  
Lower Bounds ◽  
Condition Number ◽  
Growth Rates ◽  
Separation Distance ◽  
Upper And Lower Bounds ◽  
Spectral Condition ◽  
Sharp Constants ◽  
Vandermonde Matrices

Abstract We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are “off the grid,” pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.

Global asymptotics of the Szegő–Askey polynomials

2014 ◽  
Vol 12 (06) ◽  
pp. 727-746 ◽  
Author(s):  
Y. Lin ◽  
R. Wong
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Weight Function ◽  
Unit Circle ◽  
Bessel Functions ◽  
Asymptotic Formulas ◽  
Elementary Functions ◽  
Integral Methods ◽  
Uniform Asymptotic Expansion ◽  
Global Asymptotics

The Szegő–Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann–Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369–376]. Temme's approach started from an explicit expression of the Szegő–Askey polynomials in terms of an2F1-function, and followed by integral methods.

scholarly journals Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

1999 ◽  
Vol 100 (2) ◽  
pp. 345-363 ◽  
Author(s):  
Ana Foulquié Moreno ◽  
Francisco Marcellán ◽  
K. Pan
Keyword(s):  
Asymptotic Behavior ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Sobolev Type
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