Related papers: Weak convergence of CD kernels and applications

Let m be a probability measure supported on some infinite and compact set K in the complex plane and let p_n(z) be the corresponding degree n orthonormal polynomial with positive leading coefficient. Let v_n be the normalized zero counting…

We study weak asymptotic behaviour of the Christoffel--Darboux kernel on the main diagonal corresponding to multiple orthogonal polynomials. We show that under some hypotheses the weak limit of $\tfrac{1}{n} K_n(x,x)\, d\mu$ is the same as…

In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of…

We study Nevai's condition that for orthogonal polynomials on the real line, $K_n(x,x_0)^2 K_n(x_0,x_0)^{-1} d\rho (x)\to\delta_{x_0}$ where $K_n$ is the CD kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on…

This article is a first attempt to obtain weak limit formulas for weighted means of orthogonal polynomials. For this, we introduce a new mean Nevai class that guarantees the existence of an equilibrium measure for the limit of the means. We…

Given a probability measure $\mu$ on the unit circle $\mathbb{T}$, consider the reproducing kernel $k_{\mu,n}(z_1, z_2)$ in the space of polynomials of degree at most $n-1$ with the $L^2(\mu)$-inner product. Let $u, v \in \mathbb{C}$. It is…

Let $E$ be a Banach space such that $E'$ has the Radon-Nikod\'ym property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(\mu,E)$ to a convex compactness criterion in a weaker…

We prove that in any metric space $(X,d)$ the singular integral operators {equation*} T^k_{\mu,\ve}(f)(x)=\int_{X\setminus B(x,\varepsilon)}k(x,y)f(y)d\mu (y).{equation*} converge weakly in some dense subspaces of $L^2(\mu)$ under minimal…

Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies…

We study the minimization of the energy integral $I_K(\mu) = \int_{\Omega} \int_{\Omega} K(x,y) d\mu(x) d\mu(y)$ over all Borel probability measures $\mu$, where $(\Omega,\rho)$ is a compact connected metric space and $K:\Omega^2 \to…

We show that L^2-bounded singular integral in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost…

This paper characterizes the maximum mean discrepancies (MMD) that metrize the weak convergence of probability measures for a wide class of kernels. More precisely, we prove that, on a locally compact, non-compact, Hausdorff space, the MMD…

Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d.\ non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$…

Given a sequence of polynomials $Q_n$ of degree $n$ with zeros on $[-1,1]$, we consider the triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the assumption that the sequence $\{Q_n\}$ has a weak* limiting zero…

Let $p_n(x)$ be orthogonal polynomials associated to a measure $d\mu$ of compact support in $R$. If $E\not\in supp(d\mu)$, we show there is a $\delta>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-\delta, E+\delta)$.…

Using recent results of Berman and Boucksom we show that for a non-pluripolar compact set K in C^d and an admissible weight function w=e^{-\phi} any sequence of so-called optimal measures converges weak-* to the equilibrium measure…

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$,…

Let $\Gamma \subset \mathbb C$ be a curve of class $C(2,\alpha)$. For $z_{0}$ in the unbounded component of ${\mathbb C}\setminus \Gamma$, and for $n=1,2,...$, let $\nu_n$ be a probability measure with supp$(\nu_{n})\subset \Gamma$ which…

We study optimal lower and upper bounds for Widom factors $W_{\infty,n}(K,w)$ associated with Chebyshev polynomials for the weights $w(x)=\sqrt{1+x}$ and $w(x)=\sqrt{1-x}$ on compact subsets of $[-1,1]$. We show which sets saturate these…

Denote by $\P_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$, where $D_1\in \P_{n_1}, \dots, D_k \in \P_{n_k}$ with $n_1+\cdots + n_k=n$. Partition $C\in \P_n$ according to $(n_1, \dots, n_k)$ so…