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Discrepancy densities for planar and hyperbolic zero packing
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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]

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.

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
Note

QC 20170530

Available from: 2017-05-30 Created: 2017-05-30 Last updated: 2018-05-17Bibliographically approved
In thesis
1. Random and optimal configurations in complex function theory
Open this publication in new window or tab >>Random and optimal configurations in complex function theory
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six articles spanning over several areas of mathematical analysis. The dominant theme is the study of random point processes and optimal point configurations, which may be though of as systems of charged particles with mutual repulsion. We are predominantly occupied with questions of universality, a phenomenon that appears in the study of random complex systems where seemingly unrelated microscopic laws produce systems with striking similarities in various scaling limits. In particular, we obtain a complete asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, which yields universality for the microscopic boundary behavior in the random normal matrix (RNM) model (Paper A) as well as in the case of more general interfaces for Bergman kernels (Paper B). Still in the setting of RNM ensembles, we investigate properties of scaling limits near singular points of the boundary of the spectrum, including cusps points (Paper C). We also obtain a central limit theorem for fluctuations of linear statistics in the polyanalytic Ginibre ensemble, using a new representation of the polyanalytic correlation kernel in terms of algebraic differential operators acting on the classical Ginibre kernel (Paper D). Paper E is concerned with an extremal problem for analytic polynomials, which may heuristically be interpreted as an optimal packing problem for the corresponding zeros. The last article (Paper F) concerns a different theme, namely a sharp topological transition in an Lp-analogue of classical Carleman classes for 0 < p < 1.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 46
Series
TRITA-SCI-FOU ; 2018:20
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-228092 (URN)978-91-7729-783-3 (ISBN)
Public defence
2018-06-08, Sal E2, Lindstedtsvägen 3, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20180518

Available from: 2018-05-18 Created: 2018-05-17 Last updated: 2018-05-18Bibliographically approved

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