arXiv subjects

Benedikt Steinar Magnússon

Publications and source records attributed to Benedikt Steinar Magnússon.

6 recordsLinked to original sources

arXiv2026

A Rudin-Carleson theorem with uniform approximation for manifold-valued maps

Given a closed set $E \subset \partial {\mathbb D}$ of measure zero and a continuous function $φ: E \to {\mathbb C}$, the classical Rudin-Carleson interpolation theorem states that there exists a continuous function $F : \overline {\mathbb D} \to {\mathbb C}$ that is holomorphic on ${\mathbb D}$ and satisfies $F\rvert_E = φ$. In this paper we obtain a generalisation of the Rudin-Carleson theorem for maps $φ: E \to X$ into arbitrary connected complex manifolds $X$ that combines interpolation of $φ$ on $E$ with uniform approximation on compact subsets of $\overline {\mathbb D} \setminus E$ of another given continuous map $f:\overline {\mathbb D} \to X$ that is holomorphic on ${\mathbb D}$. Under the further assumption that $X$ is an Oka manifold we obtain a corollary that combines Rudin-Carleson interpolation of a continuous map $φ: \overline {\mathbb D} \to X$ on $E$ with Runge approximation of $φ$ on a compact set $K\subset {\mathbb D}$ without any holes on which $φ$ is holomorphic.

math.CV↗

arXiv2024

A Rudin-Carleson theorem for multiply connected domains with interpolation

Using an annular version of the F. and M. Riesz theorem, we prove a generalization of the Rudin-Carleson theorem for finitely connected bounded domains. That is, for a continuous function on a closed set in the boundary of measure zero there is a holomorphic function on the domain continuous to the boundary. Furthermore, this can be done with interpolation at finitely many points in the domain. The proof relies on an annular version of the F. and M. Riesz theorem.

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 Bernstein-Walsh-Siciak theorem

We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation in $\mathbb{C}^n$ to the case where the polynomial ring $\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring $\mathcal{P}^S(\mathbb{C}^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb{R}_+^n$ with $0 \in S$ and $m = 0, 1, 2, 3, \dots$, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function.

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↗