A Novel ξ-Order Hölder Function Classes and Their Integral Transform
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Bibliographic record
Abstract
We introduce a function class \(\mathcal{H}(\xi)\) (\(\xi > 0\)) capturing tail decay and H\"older regularity. For \(h \in \mathcal{H}(\xi)\), its Fourier transform \(\mathcal{F}[h]\) inherits H\"older continuity of order \(\xi\) and essential boundedness. For \(\xi > 1\), derivatives of \(\mathcal{F}[h]\) up to order \(\lfloor \xi \rfloor\) are \(L^{\infty}\)-bounded, with fractional H\"older continuity arising from \(h\)'s decay. Our approach integrates multiscale analysis and Fourier multiplier theory, extending prior results on H\"older-Fourier correspondences. Novel integral estimates and phase cancellation methods resolve critical gaps in non-integer smoothness characterization. These results deepen the Fourier regularity analysis for non-integer \(\xi\), offering tools for harmonic analysis and pseudo-differential operators.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| 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