Fast and backward stable transforms between spherical harmonic\n expansions and bivariate Fourier series
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Bibliographic record
Abstract
A rapid transformation is derived between spherical harmonic expansions and\ntheir analogues in a bivariate Fourier series. The change of basis is described\nin two steps: firstly, expansions in normalized associated Legendre functions\nof all orders are converted to those of order zero and one; then, these\nintermediate expressions are re-expanded in trigonometric form. The first step\nproceeds with a butterfly factorization of the well-conditioned matrices of\nconnection coefficients. The second step proceeds with fast orthogonal\npolynomial transforms via hierarchically off-diagonal low-rank matrix\ndecompositions. Total pre-computation requires at best $\\mathcal{O}(n^3\\log n)$\nflops; and, asymptotically optimal execution time of $\\mathcal{O}(n^2\\log^2 n)$\nis rigorously proved via connection to Fourier integral operators.\n
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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