arXiv subjects

Konstantinos Maronikolakis

Publications and source records attributed to Konstantinos Maronikolakis.

10 recordsLinked to original sources

arXiv2026

Metric projections, zeros of optimal polynomial approximants, and some extremal problems in Hardy spaces

The well-known proof of Beurling's Theorem in the Hardy space $H^2$, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces $H^p$ for $0 < p < \infty$ are usually obtained by reduction to the $H^2$ case via inner-outer factorization of $H^p$ functions. In this paper, we instead explicitly calculate the metric projection of the unit constant function onto a shift-invariant subspace of the Hardy space $H^p$ when $1<p<\infty$. This problem is equivalent to finding the best approximation in $H^p$ of the conjugate of an inner function. In $H^2$, this approximation is always a constant, but in $H^p$, when $p\neq 2$, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff-James orthogonality and Pythagorean Inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-invariant subspaces of $H^p$, as well as the behavior of the zeros of optimal polynomial approximants in $H^p$.

math.CV↗

arXiv2025

Simultaneous Approximation by Finite Blaschke Products and Bounded Universal Functions

This paper complements the work done on simultaneous approximation results in classical Banach spaces, by focusing on approximation by finite Blaschke products. We prove the existence of a finite Blaschke product that approximates a prescribed holomorphic function bounded by 1 locally uniformly on the unit disc, and simultaneously approximates a prescribed unimodular continuous function uniformly on a compact subset of the unit circle of arclength measure 0. We also prove an analogue where the continuous function is bounded by 1 and the the approximation is achieved by an appropriate dilate of the finite Blaschke product. These results are essentially combinations of classical results of Caratheodory and Fisher on approximation by finite Blaschke products. We also give analogues for singular inner functions. Finally, we apply our results to prove the existence of bounded holomorphic functions on the unit disc that exhibit a certain universal boundary behaviour.

math.CV↗

arXiv2025

New results on universal Taylor series via weighted polynomial approximation

We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with the sup norm, while the set This improves a result of Mouze. The main ideas of the proof also allows us to construct a holomorphic function while the modulus of its non-zero Taylor coecients go to $\infty$. In passing, we complement a result by Pritsker and Varga on weighted polynomial approximation by proving that, for any compact set K with connected complement, there exists a constant $α$ K > 0 such that there exists a bounded domain G containing K such that the weighted polynomials of the form z $α$n P n , with deg(P n ) $\le$ n, are dense in H(G) for the topology of locally uniform convergence if and only if $α$ < $α$ K . Explicit computations of $α$ K are given for some simple compact sets K.

math.CV↗

arXiv2025

Quantitative incomplete polynomial approximation and frequently universal Taylor series

Let $(τ_n)_n$ be a sequence of real numbers in $(1,+\infty)$. Using potential theoretic methods, we prove quantitative results - Bernstein-Walsh type theorems - about uniform approximation by polynomials of the form $\sum_{k=\lfloor \frac{n}{τ_n} \rfloor}^na_k z^k$, on the union of two disjoint compact sets, one containing 0 and the other not. Moreover, we reveal the interplay between the compact sets and the asymptotic behaviour of the sequence $(τ_n)_n$. As applications of our results, we prove the existence of frequently universal Taylor series, with respect to the natural and the logarithmic densities, providing solutions to two problems posed by Mouze and Munnier.

math.CV↗

arXiv2024

Invariance of Abel universality under composition and applications

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all continuous functions on $K$, endowed with the supremum norm, for any proper compact subset $K$ of the unit circle. We prove that this property is invariant under composition from the left with any non-constant entire function. As an application, we show that $\mathcal{U}_A (\mathbb{D})$ is strongly-algebrable. Furthermore, we prove that Abel universality is invariant under composition from the right with an automorphism $Φ$ of $\mathbb{D}$ if and only if $Φ$ a rotation. On the other hand, we establish the existence of a subset of $\mathcal{U}_A (\mathbb{D})$ which is residual in the space of holomorphic functions on $\mathbb{D}$ and is invariant under composition from the right with any automorphism of $\mathbb{D}$.

math.CV↗

arXiv2023

Abel universal functions: boundary behaviour and Taylor polynomials

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A(\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the space of continuous functions on $K$, for any proper compact subset $K$ of the unit circle. It has been recently shown that $\mathcal{U}_A(\mathbb{D})$ is a dense $G_δ$ subset of the space of holomorphic functions on $\mathbb{D}$ endowed with the topology of local uniform convergence. In this paper, we develop further the theory of universal radial approximation by investigating the boundary behaviour of functions in $\mathcal{U}_A(\mathbb{D})$ (local growth, existence of Picard points and asymptotic values) and the convergence properties of their Taylor polynomials outside $\mathbb{D}$.

math.CV↗

arXiv2021

Universal Radial Approximation in Spaces of Analytic Functions

Recently, Charpentier showed that there exist holomorphic functions $f$ in the unit disk such that, for any proper compact subset $K$ of the unit circle, any continuous function $ϕ$ on $K$ and any compact subset $L$ of the unit disk, there exists an increasing sequence $(r_n)_{n\in\mathbb{N}}\subseteq[0,1)$ converging to 1 such that $|f(r_n(ζ-z)+z)-ϕ(ζ)|\to0$ as $n\to\infty$ uniformly for $ζ\in K$ and $z\in L$. In this paper, we give analogues of this result for the Hardy spaces $H^p(\mathbb{D}),1\leq p<\infty$. In particular, our main result implies that, if we fix a compact subset $K$ of the unit circle with zero arc length measure, then there exist functions in $H^p(\mathbb{D})$ whose radial limits can approximate every continuous function on $K$. We give similar results for the Bergman and Dirichlet spaces.

math.CV↗

arXiv2020

Universal Taylor Series in several variables depending on parameters

We establish generic existence of Universal Taylor Series on products $Ω= \prod Ω_i$ of planar simply connected domains $Ω_i$ where the universal approximation holds on products $K$ of planar compact sets with connected complements provided $K \cap Ω= \emptyset$. These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that $K \cap \overlineΩ = \emptyset$ and $\{\infty\} \cup [\mathbb{C} \setminus \overlineΩ_i]$ is connected for all $i$. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper. These universalities are topologically and algebraically generic.

math.CV↗

arXiv2020

Universal power series of Seleznev with parameters in several variables

We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products $K = \displaystyle\prod_{i = 1}^d K_i$, where $K_i \subseteq \mathbb{C}$ are compact sets and $\mathbb{C} \setminus K_i$ are connected, $i = 1, \dots, d$ and $0 \notin K$. On such $K$ the partial sums approximate uniformly any polynomial. Finally, the partial sums may be replaced by more general expressions. The phenomenon is topologically and algebraically generic.

math.CV↗

arXiv2019

An extension of the universal power series of Seleznev

We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on every compact set K not containing the origin and with connected complement. The functions b_n are assumed to be continuous and such that for every complex numbers a_0, a_1, ... , a_{n - 1}, c there exists a complex number a_n such that b_n(a_0, a_1,..., a_{n-1}, a_n) = c. This clearly covers the case of linear functions b_n.

math.CV↗