arXiv subjects

Álfheiður Edda Sigurðardóttir

Publications and source records attributed to Álfheiður Edda Sigurðardóttir.

4 recordsLinked to original sources

arXiv2025

Holomorphic approximation by polynomials with exponents restricted to a convex cone

We study the approximation of holomorphic functions of several complex variables by the ring $\mathcal{P}^S(\mathbb{C}^n)$ of polynomials whose exponents are restricted to a convex cone $\mathbb{R}_+S$ for some compact convex $S\in \mathbb{R}^n_+$. We show a version of the Runge-Oka-Weil Theorem on approximation by these subrings on compact subsets of $\mathbb{C}^{*n}$ that are convex with respect to $\mathcal{P}^S(\mathbb{C}^n)$. We show a sharper result on rotationally symmetric compact sets. The tools used are Hörmander's $L^2$-theory and Siciak-Zakharyuta functions $V^S_K$ associated to $S$. We provide a formula for $V^S_K$ when $K$ is a rotationally symmetric compact subset of $\mathbb{C}^{*n}$.

math.CV↗

arXiv2024

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results

This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the $m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb R^n_+$ is compact, convex and $0\in S$. We assume no other conditions on $S$ as these are necessary for $\mathcal P^S(\mathbb C^n)$ to be a graded polynomial ring. We study the relationship between $\mathcal P^S(\mathbb C^n)$ and the class $\mathcal L^S(\mathbb C^n)$ of global plurisubharmonic functions where the growth is determined by the logarithmic supporting function of $S$. We present properties of their respective weighted extremal functions $Φ_{K, q}^S$ and $V_{K, q}^S$ in connection with properties of $S$.

math.CV↗

arXiv2024

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Siciak-Zakharyuta theorem

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function $V_{E}$ of a subset $E$ of $\mathbb C^n$, also called a pluricomplex Green function or global exremal function of $E$, equals the logarithm of the Siciak function $Φ_E$ if $E$ is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on $E$, and the Siciak function is the upper envelope of $m$-th roots of polynomials $p$ in $\mathcal{P}_m(\mathbb C^n)$ of degree $\leq m$ such that $|p|\leq 1$ on $E$. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space ${\mathcal P}_m(\mathbb C^n)$ is replaced by ${\mathcal P}_m^S(\mathbb C^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb R^n_+$ with $0\in S$. It states that if $q$ is an admissible weight on a closed set $E$ in $\mathbb C^n$ then $V^S_{E,q}=\logΦ^S_{E,q}$ on $\mathbb C^{*n}$ if and only if the rational points in $S$ form a dense subset of $S$.

math.CV↗

arXiv2023

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Characterization of polynomials by L2-estimates

The main result of this paper is that an entire function $f$ that is in $L^2(\mathbb C^n,ψ)$ with respect to the weight $ψ(z)=2mH_S(z)+γ\log(1+|z|^2)$ is a polynomial with exponents in $m\widehat S_Γ$. Here $H_S$ is the logarithmic supporting function of a compact convex set $S\subset \mathbb R^n_+$ with $0\in S$, $γ\geq 0$ is small enough in terms of $m$, and $\widehat S_Γ$ is the hull of $S$ with respect to a certain cone $Γ$ depending on $S$, $m$ and $γ$. An example showing that in general $\widehat S_Γ$ can not be replaced by $S$ is constructed.

math.CV↗