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Orthogonal polynomials on the unit circle and chain sequences
(2013-09-01)
Szego{double acute} has shown that real orthogonal polynomials on the unit circle can be mapped to orthogonal polynomials on the interval [-1,1] by the transformation 2x=z+z-1. In the 80's and 90's Delsarte and Genin showed ...
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences
(2018-05-01)
It was shown recently that associated with a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky ...
The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials
(Wiley VCH Verlag, 2020-12-11)
Let Amp,r(n) be the best constant that fulfills the following inequality: for every m-homogeneous polynomial P(z)=∑|α|=maαzα in n complex variables, (∑|α|=m|aα|r)1/r≤Amp,r(n)supz∈Bℓnp∣∣P(z)∣∣. For every degree m, and a ...
Lamé differential equations and electrostatics
(2000-12-01)
The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A(x)y″ + 2B(x)y′ + C(x)y = 0, where A(x),B(x) and C(x) are polynomials of degree p + 1,p and p - 1, is under discussion. We ...
Matrix-Valued Gegenbauer-Type polynomials
(Springer, 2017-12)
We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν> 0. The LDU-decomposition of the weight is ...
BASIC HYPERGEOMETRIC FUNCTIONS and ORTHOGONAL LAURENT POLYNOMIALS
(Amer Mathematical Soc, 2014)
Descartes' rule of signs for polynomial systems supported on circuits
(Oxford University Press, 2017-11)
We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence ...
Bivariate orthogonal polynomials, 2D Toda lattices and Lax-type pairs
(2017-09-15)
We explore the connection between an infinite system of particles in R2 described by a bi-dimensional version of the Toda equations with the theory of orthogonal polynomials in two variables. We define a 2D Toda lattice ...
Non-Homogeneous Thermoelastic Timoshenko Systems
(Bulletin of the Brazilian Mathematical Society, New Series, 2018)
Local behavior of a polynomial near a root
(Wolfram Demonstration Project, 2011)