Continuous wavelet transform of Schwartz distributions in 𝒟′(ℝ<sup>𝑛</sup>), 𝑛 ≤ 1
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
Abstract In this paper, we extend the continuous wavelet transform to Schwartz distributions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi mathvariant="script">D</m:mi><m:mo>′</m:mo></m:msup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}^{\prime}(\mathbb{R}^{n}) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>1</m:mn></m:mrow></m:math> n\geq 1 , and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is an element <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>ψ</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \psi(x) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi mathvariant="script">D</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}(\mathbb{R}^{n}) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>1</m:mn></m:mrow></m:math> n\geq 1 , which, when integrated along each of the real axes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>X</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>X</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>X</m:mi><m:mn>3</m:mn></m:msub><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:msub><m:mi>X</m:mi><m:mi>n</m:mi></m:msub></m:mrow></m:math> X_{1},X_{2},X_{3},\ldots,X_{n} vanishes, but none of its moments <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mo largeop="true" symmetric="true">∫</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup></m:msub><m:mrow><m:mi>ψ</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mo></m:mo><m:mpadded width="+1.7pt"><m:msup><m:mi>x</m:mi><m:mi>m</m:mi></m:msup></m:mpadded><m:mo></m:mo><m:mrow><m:mo mathvariant="italic" rspace="0pt">d</m:mo><m:mi>x</m:mi></m:mrow></m:mrow></m:mrow></m:math> \int_{\mathbb{R}^{n}}\psi(x)x^{m}\,dx is zero; here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi>x</m:mi><m:mi>m</m:mi></m:msup><m:mo>=</m:mo><m:mrow><m:mpadded width="+1.7pt"><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:msub><m:mi>m</m:mi><m:mn>1</m:mn></m:msub></m:msubsup></m:mpadded><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:mn><m:msub><m:mi>m</m:mi><m:mn>2</m:mn></m:msub></m:msubsup><m:mo></m:mo><m:mi mathvariant="normal">…</m:mi><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mi>n</m:mi><m:msub><m:mi>m</m:mi><m:mi>n</m:mi></m:msub></m:msubsup></m:mrow></m:mrow></m:math> x^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mi>d</m:mi><m:mo></m:mo><m:mi>x</m:mi></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>d</m:mi><m:mo></m:mo><m:mpadded width="+1.7pt"><m:msub><m:mi>x</m:mi><m:mn>1</m:mn></m:msub></m:mpadded><m:mo></m:mo><m:mi>d</m:mi><m:mo></m:mo><m:msub><m:mi>x</m:mi><m:mn>2</m:mn></m:msub><m:mo></m:mo><m:mi mathvariant="normal">…</m:mi><m:mo></m:mo><m:mi>d</m:mi><m:mo></m:mo><m:msub><m:mi>x</m:mi><m:mi>n</m:mi></m:msub></m:mrow></m:mrow></m:math> dx=dx_{1}\,dx_{2}\ldots dx_{n} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>m</m:mi><m:mo>=</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msub><m:mi>m</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mi>n</m:mi></m:msub><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> m=(m_{1},m_{2},\ldots,m_{n}) and each of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>m</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:msub><m:mi>m</m:mi><m:mi>n</m:mi></m:msub></m:mrow></m:math> m_{1},m_{2},\ldots,m_{n} is at least 1. The set of such kernel will be denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="script">D</m:mi><m:mi>m</m:mi></m:msub><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}_{m}(\mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msubsup><m:mi mathvariant="script">D</m:mi><m:mi>F</m:mi><m:mo>′</m:mo></m:msubsup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}_{F}^{\prime}(\mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi mathvariant="script">D</m:mi><m:mo>′</m:mo></m:msup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="double-struck">R</m:mi><m:mi>n</m:mi></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> \mathcal{D}^{\prime}(\mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mfrac><m:mrow><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:mrow><m:mi mathvariant="normal">⋯</m:mi><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mi>n</m:mi><m:mn>2</m:mn></m:msubsup></m:mrow></m:mrow><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:mrow><m:mi mathvariant="normal">⋯</m:mi><m:mo></m:mo><m:msubsup><m:mi>x</m:mi><m:mi>n</m:mi><m:mn>2</m:mn></m:msubsup></m:mrow></m:mrow></m:mfrac><m:mo>=</m:mo><m:mrow><m:mn>1</m:mn><m:mo>-</m:mo><m:mfrac><m:mn>1</m:mn><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>1</m:mn><m:mn>2</m:mn></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mi>x</m:mi><m:mn>2</m:
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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.004 |
| 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) | 0.005 | 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