Admissible Majorants for Model Subspaces of <i>H</i><sup>2</sup>, Part I: Slow Winding of the Generating Inner Function
Bibliographic record
Abstract
Abstract A model subspace K ϴ of the Hardy space H 2 = H 2 (ℂ + ) for the upper half plane ℂ + is H 2 (ℂ + ) ϴ ϴ H 2 (ℂ + ) where ϴ is an inner function in ℂ + . A function ω: ⟼ [0,∞) is called an admissible majorant for K ϴ if there exists an f ∈ K ϴ , f ≢ 0, | f ( x )| ≤ ω( x ) almost everywhere on ℝ. For some (mainly meromorphic) ϴ's some parts of Adm ϴ (the set of all admissible majorants for K ϴ ) are explicitly described. These descriptions depend on the rate of growth of argϴ along ℝ. This paper is about slowly growing arguments (slower than x ). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B . A complete description of Adm B is obtained for B 's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".