The lack of a unified framework for studying

The gap

Evidence profile

Stated in the future work and cells research gap sections of the source papers, classified as general, spanning 3 journals.

Supporting evidence — 4 representative gaps

• Asymptotically Deferred Weighted Statistical Equivalence of Order α in Probability and Its Applications to Approximation Theory (2026) · Contributions to Mathematics · doi

  In this paper, we introduced and studied the notion of asymptotically deferred weighted statistical equivalence of order α in probability. By combining deferred averaging, weighting, and probabilistic asymptotic behavior, we obtained a framework that extends several existing approaches in the literature. We established the basic inclusion relations between the statistical and strong forms of equivalence and illustrated the strictness of these relations by means of a counterexample. We then applied the theory to approximation processes by establishing a Korovkin-type theorem for sequences of positive linear operators. In addition, we derived a Voronovskaya-type result describing the asymptotic behavior of the approximation error and obtained an estimate for the rate of convergence in terms of the modulus of continuity. Possible directions for future research include the study of analogous notions in ideal or lacunary settings, extensions to different classes of function spaces such as weighted spaces or Lp spaces, and multidimensional versions for functions of several variables. It would also be of interest to investigate concrete approximation processes under the present mode of convergence.

  generalstated in future workevidence 5/5
  Keywords: approximation spaces deferred weighted statistical equivalence asymptotic behavior obtained several relations processes type convergence introduced
• Statistical Korovkin-Type Theorems on Banach Function Spaces (2026) · Journal of Contemporary Applied Mathematics · doi

  In this study, statistical Korovkin-type approximation theorems were estab- lished in the setting of Banach function spaces. By introducing the shift-invariant subspace Xs, we obtained a unified framework that reduces the verification of statistical convergence for positive linear operators to their behavior on the 230 Yusuf Zeren, Migdad Ismailov, Cemil Kara¸cam minimal Korovkin test set {1, t, t2}. This extends the classical Korovkin the- ory—traditionally confined to C[0, 1] or Lp-type settings—to functional envi- ronments that may lack rearrangement invariance, smooth density, or standard modular structures. A central component of our approach is the boundedness of the Hardy– Littlewood maximal operator, which provides uniform control of the operator sequence and ensures the validity of the statistical Korovkin theorem on the entire subspace Xs. The results apply to several nonclassical spaces, including Grand Lebesgue spaces Lp)(0, 1) and Morrey-type spaces, where the classical Ko- rovkin method is not directly applicable. In these situations, our findings give new statistical approximation principles for Kantorovich-type operators and, poten- tially, for broader classes of positive linear operators dominated by maximal-type operators. Future research may examine weighted versions of Grand Lebesgue and Mor- rey spaces, statistical modular convergence in Musielak–Orlicz and variable expo- nent spaces, and the behavior of nonlinear or sublinear approximation schemes. Extensions to settings with non-doubling measures or to operators associated with fractional or localized maximal functions would further expand the reach of the statistical Korovkin framework presented here. In conclusion, the statistical approximation approach developed in this work refines classical techniques and offers a flexible methodology for connecting ab- stract operator theory with modern function space analysis. Acknowledgements. Competing Interests. The authors declare that there are no conflicts of interest associated with this publication. Authors’ Contributions. All authors have equally contributed to the con- ceptual development, analysis, and composition of the manuscript. Funding. This work was supported by the Azerbaijan Science Foundation, Grant No. AEF-MGC-2025-1(54)-20/03/1-M-03.

  generalstated in future workevidence 5/5
  Keywords: statistical spaces korovkin type operators approximation classical maximal operator authors function subspace framework convergence positive
• An Innovative Framework for Summability Methods in Measure-Theoretic Approximation (2026) · Proceedings of the National Academy of Sciences, India Section A: Physical Sciences · doi

  The lack of a comprehensive framework for summability methods in measure-theoretic approximation. - The need for a study on statistical convergence and integrability using deferred weighted summability means. - The gap in understanding the relationship between statistical convergence almost everywhere and deferred weighted statistical convergence almost everywhere.

  generalstated in cells research gapevidence 5/5
  Keywords: lack comprehensive framework summability methods measure-theoretic approximation need
• Asymptotically Deferred Weighted Statistical Equivalence of Order α in Probability and Its Applications to Approximation Theory (2026) · Contributions to Mathematics · doi

  The lack of a unified framework for studying the asymptotic behavior of random variables. - The need for a theory that combines deferred methods with weighted statistical convergence in probability.

  generalstated in cells research gapevidence 5/5
  Keywords: lack unified framework studying asymptotic behavior random variables

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