Matrix algebras with orthogonal Cauchy-like transforms and Toeplitz approximation: a negative result
Abstract
We consider newly designed matrix algebras associated with Cauchy orthogonal transforms. In this work, we are interested in checking the approximation of Toeplitz matrices using algebras of matrices, where each matrix is diagonalized by a fixed Cauchy orthogonal transform, in the spirit of fast algorithm based on preconditioning. We use the Korovkin theory for checking the Toeplitz approximability. Unfortunately and somehow surprisingly, the Korovkin test is negative, in contrast with the successful test for all the Fourier-like and trigonometric algebras, widely used in the relevant literature. More in detail, for the Cauchy algebras, the Frobenius optimal approximation fails to lead to the desired clustering, as also confirmed by ad hoc numerical experiments. It remains open to understand which kind of structured matrices can be approximated by these new algebras.
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BibTeX
@misc{fasino2026matrix,
title = {{Matrix algebras with orthogonal Cauchy-like transforms and Toeplitz approximation: a negative result}},
author = {Dario Fasino and Davide Furchì and Valerio Loi and Stefano Serra-Capizzano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16723},
eprint = {2610.16723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16723},
}
RIS
TY - UNPB
TI - Matrix algebras with orthogonal Cauchy-like transforms and Toeplitz approximation: a negative result
AU - Dario Fasino
AU - Davide Furchì
AU - Valerio Loi
AU - Stefano Serra-Capizzano
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16723
UR - https://arxiv.org/abs/2610.16723
AB - We consider newly designed matrix algebras associated with Cauchy orthogonal transforms. In this work, we are interested in checking the approximation of Toeplitz matrices using algebras of matrices, where each matrix is diagonalized by a fixed Cauchy orthogonal transform, in the spirit of fast algorithm based on preconditioning. We use the Korovkin theory for checking the Toeplitz approximability. Unfortunately and somehow surprisingly, the Korovkin test is negative, in contrast with the successful test for all the Fourier-like and trigonometric algebras, widely used in the relevant literature. More in detail, for the Cauchy algebras, the Frobenius optimal approximation fails to lead to the desired clustering, as also confirmed by ad hoc numerical experiments. It remains open to understand which kind of structured matrices can be approximated by these new algebras.
ER -