Approximation power of refinable vectors of functions
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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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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".