RATIONAL TIME-FREQUENCY GABOR FRAMES ASSOCIATED WITH PERIODIC SUBSETS OF THE REAL LINE
Bibliographic record
Abstract
For a, b > 0 and g ∈ L2(ℝ), write 𝒢(g, a, b) for the Gabor system: [Formula: see text] Let S be an aℤ-periodic measurable subset of ℝ with positive measure. It is well-known that the projection 𝒢(gχS, a, b) of a frame 𝒢(g, a, b) in L2(ℝ) onto L2(S) is a frame for L2(S). However, when ab > 1 and S ≠ ℝ, 𝒢(g, a, b) cannot be a frame in L2(ℝ) for any g ∈ L2(ℝ), while it is possible that there exists some g such that 𝒢(g, a, b) is a frame for L2(S). So the projections of Gabor frames in L2(ℝ) onto L2(S) cannot cover all Gabor frames in L2(S). This paper considers Gabor systems in L2(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L2(S) of the form 𝒢(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II).
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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 | 0.000 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".