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Search: MSC category 30C85 ( Capacity and harmonic measure in the complex plane [See also 31A15] )

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1. CMB Online first

Elmadani, Y.; Labghail, I.
Cyclicity in Dirichlet spaces
Let $\mu$ be a positive finite Borel measure on the unit circle and $\mathcal{D}(\mu)$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\mu}-$capacity, there exists a function $f\in\mathcal{D}(\mu)$ such that $f$ is cyclic (i.e., $\{p f: p $ is a polynomial$\}$ is dense in $\mathcal{D}(\mu)$), $f$ vanishes on $E$, and $f$ is uniformly continuous. Then we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in $\mathcal{D}(\mu)$.

Keywords:Dirichlet-type space, cyclic vector, capacity, strong-type inequality
Categories:47B38, 30C85, 30H05

2. CMB 2015 (vol 59 pp. 211)

Totik, Vilmos
Universality Under Szegő's Condition
This paper presents a theorem on universality on orthogonal polynomials/random matrices under a weak local condition on the weight function $w$. With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points which are Lebesgue-points both for the weight function $w$ and for $\log w$.

Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory
Categories:42C05, 60B20, 30C85, 31A15

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