On model spaces and density of functions smooth on the boundary
(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.
(Less)
- author
- Limani, Adem LU and Malman, Bartosz LU
- organization
- publishing date
- 2023
- 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}}, }