Admissible Majorants for Model Subspaces of <i>H</i><sup>2</sup>, Part II: Fast Winding of the Generating Inner Function
Bibliographic record
Abstract
Abstract This paper is a continuation of [6]. We consider the model subspaces K ϴ = H 2 ϴ ϴ H 2 of the Hardy space H 2 generated by an inner function ϴ in the upper half plane. Our main object is the class of admissible majorants for K ϴ , denoted by Adm ϴ and consisting of all functions ω defined on ℝ such that there exists an f ≠ 0, f ∈ K ϴ satisfying | f ( x )| ≤ ω( x ) almost everywhere on ℝ. Firstly, using some simple Hilbert transformtechniques, we obtain a general multiplier theorem applicable to any K ϴ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions ϴ such that the unit vector ϴ( x ) winds up fast as x grows from –∞ to ∞. In particular, we consider ϴ = B where B is a Blaschke product with “horizontal” zeros, i.e. , almost uniformly distributed in a strip parallel to and separated from ℝ. It is shown, among other things, that for any such B , any even ω decreasing on (0,∞) with a finite logarithmic integral is in Adm B (unlike the “vertical” case treated in [6]), thus generalizing (with a new proof) a classical result related to Adm exp( i σ z ), σ > 0. Some oscillating ω's in Adm B are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm exp( i σ z ), σ > 0, and to de Branges’ space ℋ( E ).
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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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 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".