相关论文: Szeg\H{o}'s theorem for Jordan arcs

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $H^2\times R$. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve homologous to…

In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N…

We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\rho$ become 'smaller' as $\rho\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function…

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha}, \alpha \in \mathbb{R} - \mathbb{Q}$). Perez-Marco showed the existence of…

We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that…

We show that a Radon measure $\mu$ in $\mathbb R^d$ which is absolutely continuous with respect to the $n$-dimensional Hausdorff measure $H^n$ is $n$-rectifiable if the so called Jones' square function is finite $\mu$-almost everywhere. The…

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega\subset\mathbb{R}^N$. It was proved by Szeg\H{o} for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains…

We study the asymptotic behavior of the Bergman orthogonal polynomials $(p_n)_{n=0}^{\infty}$ for a class of bounded simply connected domains $D$. The class is defined by the requirement that conformal maps $\varphi$ of $D$ onto the unit…

This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notion of "degree" and a more general notion that we call "logexponential degree." Specifically,…

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and…

If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at…

Let $ \Gamma < PSL_2(\mathbb{C}) $ be a Zariski dense finitely generated Kleinian group. We show all Radon measures on $ PSL_2(\mathbb{C}) / \Gamma $ which are ergodic and invariant under the action of the horospherical subgroup are either…

We study partially hyperbolic homoclinic classes of $C^1$-generic diffeomorphisms with a one-dimensional central bundle, so that the central Lyapunov exponent $\chi^c(\mu)$ is well defined for any ergodic measure $\mu$ supported on the…

Erd\"os conjectured the existence of an infinite Sidon sequence of positive integers which is also an asymptotic basis of order 3. We make progress towards this conjecture in several directions. First we prove the conjecture for all cyclic…

We consider a uniformly rectifiable set $\Gamma \subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $\Omega = \mathbb R^n \setminus \Gamma$, Guy David, Svitlana Mayboroda, and the author…

We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,\mathcal{B},\mu)$ with $\mu$ a $\sigma$-finite measure and $\mu(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known from…

In this note, we extend the well-known theorems of M. Riesz and Zygmund on conjugate functions as follows. Let $\Omega$ be a domain in $\mathbb C^n$. Suppose that $f=u+iv\in \mathcal O(\Omega)$ satisfies $v(z_0)=0$ for some $z_0\in \Omega$.…

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal…

We use nonstandard analysis to study the problem of expressing a Gaussian integral in terms of the limiting behavior of a sequence of spherical integrals. Peterson and Sengupta proved that if a Gaussian measure $\mu$ has full support on a…