On the spectral properties of the Hilbert transform operator on multi-intervals
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".