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Discrepancy densities for planar and hyperbolic zero packing
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 12, 5282-5306 p.Article in journal (Refereed) Published
Abstract [en]

We study the problem of geometric zero packing, recently introduced by Hedenmalm [7]. There are two natural densities associated with this problem: the discrepancy density ρH, given by ρH=liminfr→1−inff⁡∫D(0,r)((1−|z|2)|f(z)|−1)2dA(z)1−|z|2∫D(0,r)dA(z)1−|z|2 which measures the discrepancy in optimal approximation of (1−|z|2)−1 with the modulus of polynomials f, and its relative, the tight discrepancy density ρH ⁎, which will trivially satisfy ρH≤ρH ⁎. These densities have deep connections to the boundary behaviour of conformal mappings with k-quasiconformal extensions, which can be seen from Hedenmalm's result that the universal asymptotic variance Σ2 is related to ρH ⁎ by Σ2=1−ρH ⁎. Here we prove that in fact ρH=ρH ⁎, resolving a conjecture by Hedenmalm in the positive. The natural planar analogues ρC and ρC ⁎ to these densities make contact with work of Abrikosov on Bose–Einstein condensates. As a second result we prove that also ρC=ρC ⁎. The methods are based on Ameur, Hedenmalm and Makarov's Hörmander-type ∂¯-estimates with polynomial growth control [2]. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.

Place, publisher, year, edition, pages
Academic Press Inc. , 2017. Vol. 272, no 12, 5282-5306 p.
Keyword [en]
Asymptotic variance, Geometric zero packing, ∂¯-Estimates
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-207481DOI: 10.1016/j.jfa.2017.01.022ISI: 000400539700013Scopus ID: 2-s2.0-85012936428OAI: oai:DiVA.org:kth-207481DiVA: diva2:1108058
Note

Export Date: 22 May 2017; Article; CODEN: JFUAA; Funding details: KVA, Royal Swedish Academy of Sciences; Funding details: 2012-3122, VR, Vetenskapsrådet; Funding text: The author was supported by the Swedish Research Council, Grant no. 2012-3122, and by foundations managed by the Royal Swedish Academy of Sciences. QC 20170612

Available from: 2017-06-12 Created: 2017-06-12 Last updated: 2017-06-12Bibliographically approved

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