arXiv subjects

Bergur Snorrason

Publications and source records attributed to Bergur Snorrason.

6 recordsLinked to original sources

arXiv2026

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity

We study various regularization operators on plurisubharmonic functions that preserve Lelong classes with growth given by certain compact convex sets. The purpose is to show that the weighted Siciak-Zakharyuta functions associated with these Lelong classes are lower semicontinuous. These operators are given by integral, infimal, and supremal convolutions. Continuity properties of the logarithmic supporting function are studied and a precise description is given of when it is uniformly continuous. This gives a contradiction to published results about the Hölder continuity of these Siciak-Zakharyuta functions.

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 -- Generalized product property

A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian product. In this paper Siciak's result is generalized to the setting of Siciak-Zakharyuta functions with growth given by a compact convex set, along with discussing why this generalization does not work in the weighted setting.

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↗