An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L2(ℝ) by a Partial Sum of Its Hermite Series
Bibliographic record
Abstract
Let f be a band-limited function in L 2 ( R ) . Fix T > 0 , and suppose f ′ exists and is integrable on [ − T , T ] . This paper gives a concrete estimate of the error incurred when approximating f in the root mean square by a partial sum of its Hermite series. Specifically, we show, that for K = 2 n , n ∈ Z + , 1 2 T ∫ − T T [ f ( t ) − ( S K f ) ( t ) ] 2 d t 1 / 2 ≤ 1 + 1 K 1 2 T ∫ | t | > T f ( t ) 2 d t 1 / 2 + 1 2 T ∫ | ω | > N | f ^ ( ω ) | 2 d ω 1 / 2 + 1 K 1 2 T ∫ | t | ≤ T f N ( t ) 2 d t 1 / 2 + 1 π 1 + 1 2 K S a ( K , T ) , in which S K f is the K-th partial sum of the Hermite series of f , f ^ is the Fourier transform of f, N = 2 K + 1 + 2 K + 3 2 and f N = ( f ^ χ ( − N , N ) ) ∨ ( t ) = 1 π ∫ − ∞ ∞ sin ( N ( t − s ) ) t − s f ( s ) d s . An explicit upper bound is obtained for S a ( K , T ) .
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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 | 0.006 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".