# Variation diminishing linear time-invariant systems

arXiv: https://arxiv.org/abs/2006.10030 · HTML: https://mlsanity.com/paper/2006.10030/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2020.

## Built on by (most cited first, 1 of 1)

- [Tractable downfall of basis pursuit in structured sparse optimization](https://mlsanity.com/paper/2503.19126/) (2025)

## Abstract (arXiv)

This paper studies the variation diminishing property of k-positive linear time-invariant (LTI) systems, which map inputs with k-1 sign changes to outputs with at most the same variation. We characterize this property for the Toeplitz and Hankel operators of finite-dimensional systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of k first order positive systems. This is shown by expressing the k-positivity of a LTI system as the external positivity (that is, 1-positivity) of k compound LTI systems. Our characterization generalizes well known properties of externally positive systems (k=1) and totally positive systems (k=∞; also known as relaxation systems).