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The Factorization and Representation of Operators in the Algebra Generated by Toeplitz Operators
Stanford University, CA, USA.
Stanford University, CA, USA.
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
Stanford University, CA, USA.
1979 (English)In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 37, no 3, 467-484 p.Article in journal (Refereed) Published
##### Abstract [en]

In this paper, we study the factorization and the representation of Fredholm operators belonging to the algebra $\mathcal{R}$ generated by inversion and composition of Toeplitz integral operators. The operators in $\mathcal{R}$ have the interesting property of being close to Toeplitz (in a sense quantifiable by an integer-valued index $\alpha$) and, at the same time, of being dense in the space of arbitrary kernels. By using these properties, we derive a set of efficient algorithms (generalized fast-Cholesky and Levinson recursions) for the factorization and the inversion of arbitrary Fredholm operators. The computational burden of these algorithms depends on how close (as measured by the index $\alpha$) these operators are to being Toeplitz.

We also obtain several important representation theorems for the decomposition of operators in $\mathcal{R}$ in terms of sums of products of lower times upper triangular Toeplitz operators. These results can be used to approximate operators corresponding to noncausal and time-variant systems in terms of operators representing causal and anticausal time-invariant systems, a property that has a large number of potential applications in signal processing problems.

##### Place, publisher, year, edition, pages
SIAM , 1979. Vol. 37, no 3, 467-484 p.
##### Keyword [en]
Toeplitz operator, Fredholm operators, Factorization, Representation
##### National Category
Control Engineering
##### Identifiers
DOI: 10.1137/0137037OAI: oai:DiVA.org:liu-102126DiVA: diva2:668560
Available from: 2013-12-01 Created: 2013-12-01 Last updated: 2017-12-06

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Ljung, Lennart

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##### In the same journal
SIAM Journal on Applied Mathematics
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Control Engineering

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Cite
Citation style
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