Deferred weighted 𝒜-statistical convergence based upon the (<i>p</i>,<i>q</i>)-Lagrange polynomials and its applications to approximation theorems
Bibliographic record
Abstract
Abstract Recently, the notion of positive linear operators by means of basic (or q -) Lagrange polynomials and 𝒜 {\mathcal{A}} -statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q -Lagrange polynomials and A -statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted 𝒜 {\mathcal{A}} -statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and t 2 {t^{2}} defined on a Banach space C [ 0 , 1 ] {C[0,1]} for a sequence of (presumably new) positive linear operators based upon ( p , q ) {(p,q)} -Lagrange polynomials. Furthermore, we investigate the deferred weighted 𝒜 {\mathcal{A}} -statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.
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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 | 0.005 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| 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 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".