The Factorization and Representation of Operators in the Algebra Generated by Toeplitz Operators

• 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.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Elektroteknik och elektronik -- Reglerteknik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Electrical Engineering, Electronic Engineering, Information Engineering -- Control Engineering (hsv//eng)

Nyckelord

Toeplitz operator

Fredholm operators

Factorization

Representation

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