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On model spaces and density of functions smooth on the boundary

Limani, Adem LU and Malman, Bartosz LU (2023) In Revista Matematica Iberoamericana 39(3). p.1059-1071
Abstract

We characterize the model spaces K in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of is concentrated on a countable union of Beurling Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to any Beurling Carleson set, then even larger classes of functions, such as Holder classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in K and related spaces, the existence of a... (More)

We characterize the model spaces K in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of is concentrated on a countable union of Beurling Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to any Beurling Carleson set, then even larger classes of functions, such as Holder classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in K and related spaces, the existence of a smooth approximant is obtained through a constructive method.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
model spaces, singular inner functions, Smooth approximation
in
Revista Matematica Iberoamericana
volume
39
issue
3
pages
13 pages
publisher
EMS Publishing House
external identifiers
  • scopus:85164616777
ISSN
0213-2230
DOI
10.4171/RMI/1367
language
English
LU publication?
yes
id
aba69cad-cc90-4d77-a5ae-f9f1226348b0
date added to LUP
2023-10-04 14:58:26
date last changed
2023-10-04 14:58:26
@article{aba69cad-cc90-4d77-a5ae-f9f1226348b0,
  abstract     = {{<p>We characterize the model spaces K in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of is concentrated on a countable union of Beurling Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to any Beurling Carleson set, then even larger classes of functions, such as Holder classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in K and related spaces, the existence of a smooth approximant is obtained through a constructive method.</p>}},
  author       = {{Limani, Adem and Malman, Bartosz}},
  issn         = {{0213-2230}},
  keywords     = {{model spaces; singular inner functions; Smooth approximation}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{1059--1071}},
  publisher    = {{EMS Publishing House}},
  series       = {{Revista Matematica Iberoamericana}},
  title        = {{On model spaces and density of functions smooth on the boundary}},
  url          = {{http://dx.doi.org/10.4171/RMI/1367}},
  doi          = {{10.4171/RMI/1367}},
  volume       = {{39}},
  year         = {{2023}},
}