PreprintCC BY 4.0arXiv
The natural components of an autoregressive time series on Banach space
Abstract
We show that an autoregressive time series on Banach space can be canonically identified with a finite direct product of self-contained and self-determined natural components defined on separate subspaces by the spectral projections for each spectral point of the characteristic linear pencil. We assume the resolvent of the pencil is analytic everywhere except for the spectral points which may be poles or isolated essential singularities. Each natural component is represented as a forward, backward or outward flow fixed by an upstream boundary condition in the far distant past, the far distant future, or at a nominated initial time---depending on whether the corresponding spectral point is outside, inside, or on the unit circle.
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BibTeX
@misc{howlett2026natural,
title = {{The natural components of an autoregressive time series on Banach space}},
author = {Phil Howlett and Brendan K Beare},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17960},
eprint = {2609.17960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17960},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The natural components of an autoregressive time series on Banach space
AU - Phil Howlett
AU - Brendan K Beare
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.17960
UR - https://arxiv.org/abs/2609.17960
AB - We show that an autoregressive time series on Banach space can be canonically identified with a finite direct product of self-contained and self-determined natural components defined on separate subspaces by the spectral projections for each spectral point of the characteristic linear pencil. We assume the resolvent of the pencil is analytic everywhere except for the spectral points which may be poles or isolated essential singularities. Each natural component is represented as a forward, backward or outward flow fixed by an upstream boundary condition in the far distant past, the far distant future, or at a nominated initial time---depending on whether the corresponding spectral point is outside, inside, or on the unit circle.
ER -