Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -monotone approximation with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> is completely different from comonotone and coconvex cases. Additionally, we show that, for each function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> , the set of all monotone functions on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , and every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim sup</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:munder> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>ℙ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:munder> <mml:mfenced separators="" open="∥" close="∥"> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mfrac> </mml:mfenced> <mml:mo>≤</mml:mo> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim sup</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:munder> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>ℙ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:munder> <mml:mfenced separators="" open="∥" close="∥"> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mfrac> </mml:mfenced> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℙ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> denotes the set of algebraic polynomials of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.000 | 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