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Search: MSC category 42C05 ( Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] )

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1. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
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)

Ismail, Mourad E. H.; Obermaier, Josef
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)

Buraczewski, Dariusz; Martinez, Teresa; Torrea, José L.
Calderón--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, Calderón--Zygmund operators
Categories:42C05, 42C15frcs

4. CJM 2003 (vol 55 pp. 576)

Lukashov, A. L.; Peherstorfer, F.
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)

Duran, Antonio J.; Lopez-Rodriguez, Pedro
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

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