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September 10, 2025

Sliding Window Convergence in Real Intuitionistic Fuzzy n-Normed Linear Spaces with Some Corrections in a Recent Article

Key Points

The research identifies critical errors in prior definitions of convergence in intuitionistic fuzzy spaces, refining the theoretical framework.
By introducing new concepts like statistical limits, the study enhances the understanding of convergence in n-normed linear spaces.
Illustrative examples validate the new framework and its practical applicability in the analysis of measurable functions.
The findings resolve ambiguities in existing literature, potentially guiding future exploration in functional analysis.

Abstract

In this paper, we advance the study of sliding window convergence for a class of non-negative real-valued functions within the framework of an intuitionistic fuzzy Formula: see text-normed linear space (IFnNLS). Building on the foundational work of Connor and Savas Lacunary statistical and sliding window convergence for measurable functions, Acta Mathematica Hungarica 145 (2015) 416–432, we refine and extend their method of sliding window convergence for measurable functions, addressing significant theoretical limitations in the existing literature. We identify and rectify critical errors in a recent paper by Savas Sliding window convergence in intuitionistic fuzzy normed spaces for measurable functions, Soft Computing 26 (2022) 8299–8306, particularly in the definitions of convergence and statistical convergence, which we demonstrate to be fundamentally flawed. To bridge this research gap, we introduce novel concepts such as the statistical limit and statistical sliding window limit for functions, providing a more robust and unambiguous framework for analysis. These new definitions not only correct the deficiencies in the prior work but also offer a stronger theoretical foundation for future research. Additionally, we establish several key results in this context and validate them with illustrative examples, demonstrating the practical applicability of our approach. Our work not only resolves existing ambiguities but also opens new directions for exploring convergence in intuitionistic fuzzy spaces and beyond. This contribution marks a significant step forward in the study of sliding window convergence and its applications in functional analysis.

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