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Discrepancy densities for planar and hyperbolic zero packingPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 12, p. 5282-5306Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

ACADEMIC PRESS INC ELSEVIER SCIENCE , 2017. Vol. 272, no 12, p. 5282-5306
##### Keywords [en]

Geometric zero packing, (partial derivative)over-bar-Estimates, Asymptotic variance
##### National Category

Mathematical Analysis
##### Identifiers

URN: urn:nbn:se:kth:diva-207866DOI: 10.1016/j.jfa.2017.01.022ISI: 000400539700013Scopus ID: 2-s2.0-85012936428OAI: oai:DiVA.org:kth-207866DiVA, id: diva2:1102896
#####

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##### Note

##### In thesis

We study the problem of geometric zero packing, recently introduced by Hedenmalrn [7]. There are two natural densities associated with this problem: the discrepancy density pa, given by rho(H) = lim (r -> 1-) inf inf(f) integral(D(0,r)) ((1 - vertical bar z vertical bar(2)) vertical bar f(z)vertical bar - 1)(2) dA(z)/1 - vertical bar z vertical bar(2)/ integral(D(0,r)) dA(z)/1 - vertical bar z vertical bar(2) which measures the discrepancy in optimal approximation of (1 - vertical bar z vertical bar(2))(-1) with the modulus of polynomials f, and its relative, the tight discrepancy density rho*(H), which will trivially satisfy pH < per. 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 Sigma(2) is related to rho(H)* by Sigma(2) = 1 - rho(H)* . Here we prove that in fact rho(H) = rho(H)*, resolving a conjecture by Hedenmalm in the positive. The natural planar analogues rho(C) and rho(C)* to these densities make contact with work of Abrikosov on Bose Einstein condensates. As a second result we prove that also rho(C) = rho(C)*. The methods are based on Ameur, Hedenmalm and Makarov's Hormander-type <(partial derivative)over bar>-estimates with polynomial growth control [2]. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.

QC 20170530

Available from: 2017-05-30 Created: 2017-05-30 Last updated: 2018-05-17Bibliographically approved1. Random and optimal configurations in complex function theory$(function(){PrimeFaces.cw("OverlayPanel","overlay1206742",{id:"formSmash:j_idt1523:0:j_idt1528",widgetVar:"overlay1206742",target:"formSmash:j_idt1523:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

doi
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