# A Generalized Nyquist-Shannon Sampling Theorem Using the Koopman Operator

arXiv: https://arxiv.org/abs/2303.01927 · HTML: https://mlsanity.com/paper/2303.01927/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2023.

## Built on by (most cited first, 3 of 3)

- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [Rethinking Chronological Causal Discovery with Signal Processing](https://mlsanity.com/paper/2602.19903/) (2026)
- [A Novel Unified Extended Matrix for Graph Signal Processing: Theory and Application](https://mlsanity.com/paper/2508.16633/) (2025)

## Abstract (arXiv)

In the field of signal processing, the sampling theorem plays a fundamental role for signal reconstruction as it bridges the gap between analog and digital signals. Following the celebrated Nyquist-Shannon sampling theorem, generalizing the sampling theorem to non-band-limited signals remains a major challenge. In this work, a generalized sampling theorem, which builds upon the Koopman operator, is proposed for signals in a generator-bounded space. It naturally extends the Nyquist-Shannon sampling theorem in that: 1) for band-limited signals, the lower bounds of the sampling frequency and the reconstruction formulas given by these two theorems are exactly the same; 2) the Koopman operator-based sampling theorem can also provide a finite bound of the sampling frequency and a reconstruction formula for certain types of non-band-limited signals, which cannot be addressed by Nyquist-Shannon sampling theorem. These non-band-limited signals include, but are not limited to, the inverse Laplace transform with limit imaginary interval of integration, and linear combinations of complex exponential functions. Furthermore, the Koopman operator-based reconstruction method is supported by theoretical results on its convergence. This method is illustrated numerically through several examples, demonstrating its robustness against low sampling frequencies.