# Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2102.13266 · HTML: https://mlsanity.com/paper/2102.13266/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 67 references that are in the map.
- First posted on arXiv 26 Feb 2021.

## Builds on (most cited first, 5 of 5)

- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Machine Learning of Space-Fractional Differential Equations](https://mlsanity.com/paper/1808.00931/) (2018)
- [The Occupation Kernel Method for Nonlinear System Identification](https://mlsanity.com/paper/1909.11792/) (2019)
- [Technical Report: Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures](https://mlsanity.com/paper/1305.7484/) (2013)
- [Invariant Sets in Quasiperiodically Forced Dynamical Systems](https://mlsanity.com/paper/1808.08340/) (2018)

## Abstract (arXiv)

This manuscript gives a theoretical framework for a new Hilbert space of functions, the so called occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex numbers. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented, and the details of the finite rank representations are given. Significantly, despite the added theoretical content through the OKHS formulation, the resultant computations only differ slightly from that of occupation kernel DMD methods for integer order systems posed over RKHSs.