Admissible Majorants for Model Subspaces of <i>H</i><sup>2</sup>, Part II: Fast Winding of the Generating Inner Function
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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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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 | 0.002 | 0.001 |
| 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 | 0.000 |
| 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