A Generalized Nyquist-Shannon Sampling Theorem Using the Koopman Operator
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Bibliographic record
Abstract
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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it