UNCONDITIONAL CONVERGENCE AND UNCONDITIONAL BASES IN HARDY SPACES
Why this work is in the frame
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Bibliographic record
Abstract
This paper is devoted to a study of unconditional convergence of series in the Hardy space H 1 (ℝ s ) and unconditional bases for H 1 (ℝ s ). First, we use quasi-projection operators from approximation theory to give a very general criterion for unconditional convergence in H 1 . Second, we prove that a system of wavelets forms an unconditional basis of the Hardy space H 1 , provided the dual wavelet lies in a Lipschitz space of positive order. In particular, for H 1 (ℝ) we construct an unconditional basis consisting of piecewise constant functions. Third, we demonstrate that our conditions for unconditional bases are sharp by showing that, if the dual refinable function is the characteristic function of the interval [0, 1), then the corresponding system of wavelets does not form an unconditional basis for H 1 (ℝ), even though the wavelet itself could have arbitrarily high smoothness.
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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 | 0.000 | 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) | 0.001 | 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