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1. CJM 2012 (vol 65 pp. 600)
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials We establish asymptotics for Christoffel functions associated with
multivariate orthogonal polynomials. The underlying measures are assumed to
be regular on a suitable domain - in particular this is true if they are
positive a.e. on a compact set that admits analytic parametrization. As a
consequence, we obtain asymptotics for Christoffel functions for measures on
the ball and simplex, under far more general conditions than previously
known. As another consequence, we establish universality type limits in the
bulk in a variety of settings.
Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions Categories:42C05, 42C99, 42B05, 60B20 |
2. CJM 2010 (vol 63 pp. 181)
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the Askey--Wilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$-ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$-Jacobi polynomials is given.
Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients Categories:33D45, 42C05 |
3. CJM 2007 (vol 59 pp. 1223)
CalderÃ³n--Zygmund Operators Associated to Ultraspherical Expansions We define the higher order Riesz transforms and the Littlewood--Paley
$g$-function
associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta
f'(\theta)+\l^2f(\theta)$. We prove that these operators are
Calder\'{o}n--Zygmund operators in the homogeneous type space
$((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted,
$H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.
Keywords:ultraspherical polynomials, CalderÃ³n--Zygmund operators Categories:42C05, 42C15frcs |
4. CJM 2003 (vol 55 pp. 576)
Automorphic Orthogonal and Extremal Polynomials It is well known that many polynomials which solve extremal problems
on a single interval as the Chebyshev or the Bernstein-Szeg\"o
polynomials can be represented by trigonometric functions and their
inverses. On two intervals one has elliptic instead of trigonometric
functions. In this paper we show that the counterparts of the Chebyshev
and Bernstein-Szeg\"o polynomials for several intervals can be represented
with the help of automorphic functions, so-called Schottky-Burnside
functions. Based on this representation and using the Schottky-Burnside
automorphic functions as a tool several extremal properties of such
polynomials as orthogonality properties, extremal properties with
respect to the maximum norm, behaviour of zeros and recurrence
coefficients {\it etc.} are derived.
Categories:42C05, 30F35, 31A15, 41A21, 41A50 |
5. CJM 1997 (vol 49 pp. 708)
Density questions for the truncated matrix moment problem For a truncated matrix moment problem, we describe in detail
the set of positive definite matrices of measures $\mu$ in $V_{2n}$ (this is
the set of solutions of the problem of degree $2n$) for which the polynomials
up to degree $n$ are dense in the corresponding space ${\cal L}^2(\mu)$.
These matrices of measures are exactly the extremal measures of the set $V_n$.
Categories:42C05, 44A60 |