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On the Structure of Positive Semi-Definite Finite Rank General Domain Hankel and Toeplitz Operators in Several Variables

Andersson, Fredrik LU and Carlsson, Marcus LU (2017) In Complex Analysis and Operator Theory 11(4). p.755-784
Abstract

Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published, that characterize the generating functions that give rise to finite rank multidimensional Hankel and Toeplitz type operators defined on general domains. In this paper we study how the additional assumption of positive semi-definite affects the characterization of the corresponding generating functions. We show that these theorems become particularly transparent in the continuous setting, by providing elegant if-and-only-if statements connecting the rank with sums of exponential functions. We also discuss how these operators can be discretized, giving rise to an interesting class of structured matrices that inherit desirable... (More)

Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published, that characterize the generating functions that give rise to finite rank multidimensional Hankel and Toeplitz type operators defined on general domains. In this paper we study how the additional assumption of positive semi-definite affects the characterization of the corresponding generating functions. We show that these theorems become particularly transparent in the continuous setting, by providing elegant if-and-only-if statements connecting the rank with sums of exponential functions. We also discuss how these operators can be discretized, giving rise to an interesting class of structured matrices that inherit desirable properties from their continuous analogs. In particular we describe how the continuous Kronecker theorem also applies to these structured matrices, given sufficient sampling. We also provide a new proof for the Carathéodory-Fejér theorem for block Toeplitz matrices, based on tools from tensor algebra.

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author
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type
Contribution to journal
publication status
published
subject
keywords
Carathéodory-Fejér theorem, Finite rank, Hankel, Kronecker theorem, Sums of exponentials, Toeplitz
in
Complex Analysis and Operator Theory
volume
11
issue
4
pages
30 pages
publisher
Springer
external identifiers
  • wos:000398768500003
  • scopus:84992699610
ISSN
1661-8254
DOI
10.1007/s11785-016-0596-6
language
English
LU publication?
yes
id
cdb1c7d9-cc8d-4567-a581-84d8d4f19235
date added to LUP
2016-11-14 11:48:24
date last changed
2024-10-05 05:35:00
@article{cdb1c7d9-cc8d-4567-a581-84d8d4f19235,
  abstract     = {{<p>Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published, that characterize the generating functions that give rise to finite rank multidimensional Hankel and Toeplitz type operators defined on general domains. In this paper we study how the additional assumption of positive semi-definite affects the characterization of the corresponding generating functions. We show that these theorems become particularly transparent in the continuous setting, by providing elegant if-and-only-if statements connecting the rank with sums of exponential functions. We also discuss how these operators can be discretized, giving rise to an interesting class of structured matrices that inherit desirable properties from their continuous analogs. In particular we describe how the continuous Kronecker theorem also applies to these structured matrices, given sufficient sampling. We also provide a new proof for the Carathéodory-Fejér theorem for block Toeplitz matrices, based on tools from tensor algebra.</p>}},
  author       = {{Andersson, Fredrik and Carlsson, Marcus}},
  issn         = {{1661-8254}},
  keywords     = {{Carathéodory-Fejér theorem; Finite rank; Hankel; Kronecker theorem; Sums of exponentials; Toeplitz}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{755--784}},
  publisher    = {{Springer}},
  series       = {{Complex Analysis and Operator Theory}},
  title        = {{On the Structure of Positive Semi-Definite Finite Rank General Domain Hankel and Toeplitz Operators in Several Variables}},
  url          = {{http://dx.doi.org/10.1007/s11785-016-0596-6}},
  doi          = {{10.1007/s11785-016-0596-6}},
  volume       = {{11}},
  year         = {{2017}},
}