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

2023/05/11 by Benedikt Steinar Magnússon, Álfheiður Edda Sigurðardóttir, Magnússon, Benedikt Steinar +5
Mathematics · #32A15 #32U15 #32U35 (Primarey) 32A08 #32W50 (Secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2305.06847

openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The main result of this paper is that an entire function f that is in L²(\mathbb Cⁿ,ψ) with respect to the weight ψ(z)=2mH_S(z)+γlog(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⊂ \mathbb Rⁿ₊ with 0∈ S, γ≥ 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.

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