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Artur Nicolau

Publications and source records attributed to Artur Nicolau.

At least 19 recordsLinked to original sources
arXiv2026

From local to global asymptotic behaviour of orthogonal polynomials

Let $\{ϕ^*_n\}$ be the sequence of reflected orthogonal polynomials on the unit circle $\partial \mathbb{D}$ generated by a measure $μ$ of Szegő class, and let $D_μ$ be the Szegő function of $μ$. We prove the uniform Cesàro asymptotics $$ \sup_{z \in Γ_ζ}\Biggl(\frac{1}{n}\sum_{k = 0}^{n-1}\Bigl||ϕ_k^*(z) D_μ(z)|^2 - 1\Bigr|\Biggr) \to 0, \qquad n \to \infty, $$ for almost all Stolz angles $Γ_ζ$, $ζ\in \partial \mathbb{D}$. This extends a well-known asymptotic result of Máté, Nevai, and Totik (1991) from the local scale $O(1/n)$ near $\partial \mathbb{D}$ to the global scale $O(1)$. We also study asymptotic behavior of arguments of orthogonal polynomials and extend a classical theorem due to Grenander and Szegő using a new technique. As an application, we derive global asymptotic results for polynomial reproducing kernels under various assumptions on the orthogonality measure.

math.CA↗
arXiv2026

The Law of the iterated logaritm for smooth functions

A version of the Law of the Iterated Logarithm for smooth functions in the upper-half space is proved. As a consequence, we show that certain size conditions on the gradient and the gradient of the laplacian of a smooth function, lead to self-improvement growth properties. The results are applied in situations where harmonicity is not present.

math.CA↗
arXiv2026

Contractive analytic self-mappings of the disc

Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their inner-outer factorization. In addition, we show that contractive inner functions can be described in terms of a certain mixing property of its boundary values. We also present other results on the boundary behavior of contractive inner functions.

math.CV↗
arXiv2025

Carleson Measures, Vanishing Mean Oscillation and Critical Points

Given a finite positive Borel measure $μ$ in the open unit disc of the complex plane, we construct a bounded outer function $E$ whose boundary values have vanishing mean oscillation such that $|E| μ$ is a vanishing Carleson measure. As an application it is shown that given any function in a Hardy space, there exists a bounded analytic function in the unit disc whose boundary values have vanishing mean oscillation, with the same critical points and multiplicities.

math.CV↗
arXiv2025

Analytic mappings of the unit disk with bounded compression

In this paper, we study analytic self-maps of the unit disk for which the hyperbolic diameters of the images of hyperbolic balls of radius 1 are uniformly bounded below. We give several characterizations of such maps involving the behaviour along geodesic rays, Aleksandrov-Clark measures, zero sets and critical sets.

math.CV↗
arXiv2025

Inner Functions, Möbius Distortion and Angular Derivatives

We prove that an inner function has finite $\mathcal{L} (p)$-entropy if and only if its accumulated Möbius distortion is in $L^p$, $0<p<\infty$. We also study the support of the positive singular measures such that their corresponding singular inner functions have finite $\mathcal{L} (p)$-entropy.

math.CV↗
arXiv2025

Shift invariant subspaces in the Bloch space

We consider weak-star closed invariant subspaces of the shift operator in the classical Bloch space. We prove that any bounded analytic function decomposes into two factors, one which is cyclic and another one generating a proper shift invariant subspace, satisfying a permanence property, which in a certain way is opposite to cyclicity. Singular inner functions play the crucial role in this decomposition. We show in several different ways that the description of shift invariant subspaces generated by inner functions in the Bloch spaces deviates substantially from the corresponding description in the Bergman spaces, provided by the celebrated Korenblum and Roberts Theorem. Furthermore, the relationship between invertibility and cyclicity is also investigated and we provide an invertible function in the Bloch space which is not cyclic therein. Our results answer several open questions stated in the early nineties.

math.FA↗
arXiv2025

Sharp Invertibility in Quotient Algebras of $H^\infty$

We consider inner functions $Θ$ with the zero set $\mathcal Z(Θ)$ such that the quotient algebra $H^\infty / ΘH^\infty$ satisfies the Strong Invertibility Property (SIP), that is for every $\varepsilon>0$ there exists $δ>0$ such that the conditions $f \in H^\infty$, $\|[f]\|_{H^\infty/ ΘH^\infty}=1$, $\inf_{\mathcal Z(Θ)} |f| \ge 1-δ$ imply that $[f]$ is invertible in $H^\infty / ΘH^\infty$ and $\| 1/ [f] \|_{H^\infty/ ΘH^\infty}\le 1+\varepsilon$. We prove that the SIP is equivalent to the maximal asymptotic growth of $Θ$ away from its zero set. We also describe inner functions satisfying the SIP in terms of the narrowness of their sublevel sets and relate the SIP to the Weak Embedding Property introduced by P.Gorkin, R.Mortini, and N.Nikolski as well as to inner functions whose Frostman shifts are Carleson--Newman Blaschke products. We finally study divisors of inner functions satisfying the SIP. We describe geometrically the zero set of inner functions such that all its divisors satisfy the SIP. We also prove that a closed subset $E$ of the unit circle is of finite entropy if and only if any singular inner function associated to a singular measure supported on $E$ is a divisor of an inner function satisfying the SIP.

math.CV↗
arXiv2024

Analytic mappings of the unit disk which almost preserve hyperbolic area

In this paper, we study analytic self-maps of the unit disk which distort hyperbolic area of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, Möbius distortion, the distribution of critical points and Aleksandrov-Clark measures. We also study Lyapunov exponents of their Aleksandrov-Clark measures.

math.CV↗
arXiv2024

The Central Limit Theorem for inner functions II

A sharp version of the Central Limit Theorem for linear combinations of iterates of an inner function is proved. The authors previously showed this result assuming a suboptimal condition on the coefficients of the linear combination. Here we explain a variation of the original argument which leads to the sharp result. We also review the steps of the proof as well as the main technical tool, which is Aleksandrov Desintegration Theorem for Aleksandrov-Clark measures.

math.CV↗
arXiv2023

Beurling-Carleson sets, inner functions and a semi-linear equation

Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling-Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling-Carleson sets. For a positive singular measure $μ$ on the unit circle, let $S_μ$ denote the singular inner function with singular measure $μ$. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle such as membership of $S'_μ$ in the Nevanlinna class $\mathcal N$, area conditions on level sets of $S_μ$ and wepability. It was known that each of these properties holds for measures concentrated on Beurling-Carleson sets. We show that each of these properties implies that $μ$ lives on a countable union of Beurling-Carleson sets. We also describe partial relations involving the membership of $S'_μ$ in the Hardy space $H^p$, membership of $S_μ$ in the Besov space $B^p$ and $(1-p)$-Beurling-Carleson sets and give a number of examples which show that our results are optimal. Finally, we show that measures that live on countable unions of $α$-Beurling-Carleson sets are almost in bijection with nearly-maximal solutions of $Δu = u^p \cdot χ_{u > 0}$ when $p > 3$ and $α= \frac{p-3}{p-1}$.

math.CV↗
arXiv2022

One-component bounded functions

Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the corresponding de Branges-Rovnyak spaces and the last one concerns the associated Clark measure.

math.CV↗
arXiv2021

Iterates of Blaschke products and Peano curves

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Given a sequence $\{a_{n}\}$ of complex numbers consider $F= \sum a_n f^{n}$. If $\{a_n\}$ tends to $0$ but $\sum |a_n| = \infty$, we prove that for any complex number $w$ there exists a point $ξ$ in the unit circle such that $\sum a_{n}f^{n}(ξ)$ converges and its sum is $w$. If $\sum |a_n| < \infty$ and the convergence is slow enough in a certain precise sense, then the image of the unit circle by $F$ has a non empty interior. The proofs are based on inductive constructions which use the beautiful interplay between the dynamics of $f$ as a selfmapping of the unit circle and those as a selfmapping of the unit disc.

math.CA↗
arXiv2021

Convergence of linear combinations of iterates of an inner function

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Let $\{a_{n}\}$ be a sequence of complex numbers. We prove that the series $\sum a_{n}f^{n}(ξ)$ converges at almost every point $ξ$ of the unit circle if and only if $\sum |a_n|^2 < \infty$. The main step in the proof is to show that under this assumption, the function $F= \sum a_n f^n$ has bounded mean oscillation. We also prove that $F$ is bounded on the unit disc if and only if $\sum |a_n| < \infty$. Finally we describe the sequences of coefficients $\{a_n \}$ such that $F$ belongs to other classical function spaces, as the disc algebra and the Dirichlet class.

math.CV↗
arXiv2020

Bloch functions and Bekollé-Bonami weights

We study analogues of well-known relationships between Muckenhoupt weights and $BMO$ in the setting of Bekollé-Bonami weights. For Bekollé-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekollé-Bonami weight, which in particular reveals an intimate connection between Bekollé-Bonami weights and Bloch functions. On the open problem of characterizing the closure of bounded analytic functions in the Bloch space, we provide a counter-example to a related recent conjecture. This shed light into the difficulty of preserving harmonicity in approximation problems in norms equivalent to the Bloch norm. Finally, we apply our results to study certain spectral properties of Cesaró operators.

math.CV↗
arXiv2020

A Central Limit Theorem for Inner Functions

A Central Limit Theorem for linear combinations of iterates of an inner function is proved. The main technical tool is Aleksandrov Desintegration Theorem for Aleksandrov-Clark measures.

math.CV↗
arXiv2020

A Characterization of One-component Inner Functions

We present a characterization of one-component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by J. Cima and R. Mortini. In particular we prove that for any inner function $Θ$ whose singular set has measure zero, one can find a Blaschke product $B$ such that $ΘB$ is one-component. We also obtain a characterization of one-component singular inner functions which is used to produce examples of discrete and continuous one-component singular inner functions.

math.CA↗