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

Abstract

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.