Approximation power of refinable vectors of functions
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Bibliographic record
Abstract
In this paper we survey recent results on approximation power of refinable vectors of functions. Let Φ = (φ1, . . . , φr) be an r × 1 vector of compactly supported functions in Lp(IR) (1 ≤ p ≤ ∞). The first part of this paper is devoted to an investigation of approximation power of S(Φ), the shift-invariant space generated from Φ. We review results on characterizations of the approximation order of S(Φ) and describe approximation schemes that achieve the optimal approximation order. We also give a self-contained treatment of various equivalent forms of the Strang-Fix conditions. We say that Φ is refinable if Φ = ∑ α∈Z s a(α)Φ(M · − α), where M is an expansive s× s integer matrix, and the refinement mask a is finitely supported. The second part of this paper is dedicated to a study of accuracy of Φ. We review results on characterizations of the accuracy of Φ in terms of the mask in both time and frequency domains. We also discuss the relationship between the accuracy of Φ and the sum rules associated with the mask. Examples are provided to illustrate the general theory. † Supported in part by NSERC Canada under Grant OGP 121336 Approximation Power of Refinable Vectors of Functions §
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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) | 0.001 | 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