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<p>This article delves into Korovkin-type theorems in Banach function spaces as established by Zeren et al. (Positivity 26:28, 2022). We prove that in this theorem, the positivity of the operators is not a necessary requirement and provides an example of a sequence of non-positive operators where it is applicable. Under the assumption of positivity, we establish an operator version of the result. Additionally, we derive a quantitative form of the result using the modulus of continuity. We apply the result to examples, such as Lebesgue space, Weighted Lebesgue space, Grand Lebesgue space, etc. Furthermore, we present numerical illustrations for specific cases.</p>

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Revisiting Korovkin-Type Theorems in Banach Function Spaces

  • 期刊论文

  • 发表日期：

    2025年1月6日

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  • V. B. Kiran Kumar,
  • P. C. Vinaya

摘要

This article delves into Korovkin-type theorems in Banach function spaces as established by Zeren et al. (Positivity 26:28, 2022). We prove that in this theorem, the positivity of the operators is not a necessary requirement and provides an example of a sequence of non-positive operators where it is applicable. Under the assumption of positivity, we establish an operator version of the result. Additionally, we derive a quantitative form of the result using the modulus of continuity. We apply the result to examples, such as Lebesgue space, Weighted Lebesgue space, Grand Lebesgue space, etc. Furthermore, we present numerical illustrations for specific cases.