Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
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: 000400539700013ScopusID: 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

Open Access in DiVA

No full text

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Wennman, Aron
By organisation
Mathematics (Div.)
In the same journal
Journal of Functional Analysis
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

Altmetric score

CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf