Interpolation and duality in algebras of multipliers on the ball
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Bibliographic record
Abstract
We study the multiplier algebras A(\mathcal{H}) obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces \mathcal{H} on the ball \mathbb{B}_d of \mathbb{C}^d . Our results apply, in particular, to the Drury–Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of A(\mathcal H) in terms of the complementary bands of Henkin and totally singular measures for \operatorname{Mult}(\mathcal{H}) . This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact \operatorname{Mult}(\mathcal{H}) -totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is \operatorname{Mult}(\mathcal{H}) -totally null.
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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 | 0.004 | 0.002 |
| 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 | 0.001 |
| 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