Norming Sets and Related Remez-type Inequalities
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Bibliographic record
Abstract
The classical Remez inequality bounds the maximum of the absolute value of a real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available. Still, given a subset $Z\subset [-1,1]^n\subset {\mathbb R}^n$ it is not easy to determine whether it is ${\mathcal P}_d({\mathbb R}^n)$-norming (here ${\mathcal P}_d({\mathbb R}^n)$ is the space of real polynomials of degree at most $d$ on ${\mathbb R}^n$), i.e. satisfies a Remez-type inequality: $\sup_{[-1,1]^n}|P|\le C\sup_{Z}|P|$ for all $P\in {\mathcal P}_d({\mathbb R}^n)$ with $C$ independent of $P$. (Although ${\mathcal P}_d({\mathbb R}^n)$-norming sets are exactly those not contained in any algebraic hypersurface of degree $d$ in ${\mathbb R}^n$, there are many apparently unrelated reasons for $Z \subset [-1,1]^n$ to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces $V$ of continuous functions on $[-1,1]^n$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for $Z$ to be $V$-norming, partly known, partly new, restricting ourselves to the simplest non-trivial examples. Next, we extend the Turan-Nazarov inequality for exponential polynomials to several variables, and on this base prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants $N_{V}(Z)$ in the Remez-type inequalities for $V$, as the function of the set $Z$, showing that it is Lipschitz in the Hausdorff metric.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.002 | 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