TY - JOUR

T1 - Nevanlinna domains with large boundaries

AU - Belov, Yurii

AU - Borichev, Alexander

AU - Fedorovskiy, Konstantin

PY - 2019/10/15

Y1 - 2019/10/15

N2 - Nevanlinna domains are an important class of bounded simply connected domains in the complex plane; they are images of the unit disc under mappings by univalent functions belonging to model spaces (i.e. the subspaces of the Hardy space H2 invariant with respect to the backward shift operator). Nevanlinna domains play a crucial role in recent progress in problems of uniform approximation of functions on compact sets in C by polyanalytic polynomials and polyanalytic rational functions. We give a complete solution to the following problem posed in the early 2000s: how large (in the sense of dimension) can be the boundaries of Nevanlinna domains? We establish the existence of Nevanlinna domains with large boundaries. In particular, these domains can have boundaries of positive planar measure. The sets of accessible points can be of any Hausdorff dimension between 1 and 2. As a quantitative counterpart of these results, we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles.

AB - Nevanlinna domains are an important class of bounded simply connected domains in the complex plane; they are images of the unit disc under mappings by univalent functions belonging to model spaces (i.e. the subspaces of the Hardy space H2 invariant with respect to the backward shift operator). Nevanlinna domains play a crucial role in recent progress in problems of uniform approximation of functions on compact sets in C by polyanalytic polynomials and polyanalytic rational functions. We give a complete solution to the following problem posed in the early 2000s: how large (in the sense of dimension) can be the boundaries of Nevanlinna domains? We establish the existence of Nevanlinna domains with large boundaries. In particular, these domains can have boundaries of positive planar measure. The sets of accessible points can be of any Hausdorff dimension between 1 and 2. As a quantitative counterpart of these results, we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles.

KW - Hausdorff dimension

KW - Model space K

KW - Nevanlinna domain

KW - Univalent rational function

KW - Model space KΘ

KW - SPACES

KW - APPROXIMABILITY

KW - UNIFORM APPROXIMATION

KW - COMPACT-SETS

KW - POLYNOMIAL APPROXIMATIONS

KW - REGULARITY

KW - Model space K-circle minus

KW - EXAMPLE

KW - SPECTRUM

UR - http://www.scopus.com/inward/record.url?scp=85060110545&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/nevanlinna-domains-large-boundaries

U2 - 10.1016/j.jfa.2018.12.015

DO - 10.1016/j.jfa.2018.12.015

M3 - Article

AN - SCOPUS:85060110545

VL - 277

SP - 2617

EP - 2643

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -