A connection between Schur multiplication and Fourier interpolation. II
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Bibliographic record
Abstract
Abstract Given m × n matrices A = [ a jk ] and B = [ b jk ], their Schur product is the m × n matrix A ○ B = [ a jk b jk ]. For any matrix T , define ‖ T‖ S = max X ≠ O ‖ T ○ X ‖/‖ X ‖ (where ‖·‖ denotes the usual matrix norm). For any complex (2 n – 1)‐tuple μ = ( μ – n +1 , μ – n +2 , …, μ n –1 ), let T μ be the Hankel matrix [ μ – n + j + k –1 ] j,k and define 𝔅 μ = { f ∈ L 1 [–π, π] : f̂ (2 j ) = μ j for – n + 1 ≤ j ≤ n – 1} . It is known that ‖ T μ ‖ S ≤ inf ‖ f ‖ 1 . When equality holds, we say T μ is distinguished. Suppose now that μ j ∈ ℝ for all j and hence that T μ is hermitian. Then there is a real n × n hermitian unitary X and a real unit vector y such that 〈( T μ ○ X ) y , y 〉 = ‖ T μ ‖ S . We call such a pair a norming pair for T μ . In this paper, we study norming pairs for real Hankel matrices. Specifically, we characterize the pairs that norm some distinguished Schur multiplier T μ . We do this by giving necessary and suf.cient conditions for ( X , y ) to be a norming pair in the n ‐dimensional case. We then consider the 2‐ and 3‐dimensional cases and obtain further results. These include a new and simpler proof that all real 2 × 2 Hankel matrices are distinguished, and the identi.cation of new classes of 3 × 3 distinguished matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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