On the spectral properties of the Hilbert transform operator on multi-intervals
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Bibliographic record
Abstract
Let J,E\subset\mathbb{R} be two multi-intervals with non-intersecting interiors. Consider the operator A\colon L^2( J )\to L^2(E),\quad (Af)(x) = \frac 1\pi\int_J \frac {f(y) d y}{{y-x}}, and let A^\dagger be its adjoint. We introduce a self-adjoint operator \mathscr K acting on L^2(E)\oplus L^2(J) , whose off-diagonal blocks consist of A and A^\dagger . In this paper we study the spectral properties of \mathscr K and the operators A^\dagger A and A A^\dagger . Our main tool is to obtain the resolvent of \mathscr K , which is denoted by \mathscr R , using an appropriate Riemann–Hilbert problem, and then compute the jump and poles of \mathscr R in the spectral parameter \lambda . We show that the spectrum of \mathscr K has an absolutely continuous component [0,1] if and only if J and E have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of \mathscr K consists only of eigenvalues and 0 . If there are common endpoints, then \mathscr K may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at 0 . In all cases, \mathscr K does not have a singular continuous spectrum. The spectral properties of A^\dagger A and A A^\dagger , which are very similar to those of \mathscr K , are obtained as well.
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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