On the Generalized Weighted Statistical Convergence

Year 2024, , 1310 - 1314, 15.11.2024

Çiğdem Bektaş , Erdal Bayram

https://doi.org/10.34248/bsengineering.1553162

Abstract

Statistical convergence and summability represent a significant generalization of traditional convergence for sequences of real or complex values, allowing for a broader interpretation of convergence phenomena. This concept has been extensively examined by numerous researchers using various mathematical tools and applied to different mathematical structures over time, revealing its relevance across multiple disciplines. In the present study, a generalized definition of the concepts of statistical convergence and summability, termed (△_v^m )_u-generalized weighted statistical convergence and (△_v^m )_u-generalized weighted by [¯N_t ]-summability for real sequences, is introduced using the weighted density and generalized difference operator. Based on this definition, several fundamental properties and inclusion results, obtained by differentiating the components used in the definitions, are provided.

Keywords

Generalized difference sequence, Weighted density, Weighted statistical convergence, Weighted summability

References

• Barlak D. 2020. Statistical convergence of order β for (λ,μ) double sequences of fuzzy numbers, 39(5): 6949-6954.
• Bektaş ÇA, Çolak R. 2005. On some generalized difference sequence spaces. Thai J Math, 3(1): 83-98.
• Belen C, Mohiuddine SA. 2013. Generalized weighted statistical convergence and application. Appl Math Computat, 219(18): 9821-9826.
• Braha NL, Srivastava HM, Et M. 2021. Some weighted statistical convergence and associated Korovkin and Voronovskaya type theorems. J App Math Comput, 65: 429-450.
• Connor JS. 1988. The statistical and strong p-Cesaro convergence of sequences. Analysis, 8: 47-63.
• Et M, Çolak R. 1995. On generalized difference sequence spaces. Soochow J Math, 21(4): 377-386.
• Et M, Esi A. 2000. On Köthe-Toeplitz duals of generalized difference sequence spaces. Bull Malaysian Math Sci Soc, 23: 25-32.
• Et M, Nuray F. 2001. △^m-Statistical Convergence. Indian J Pure Appl Math, 32(6): 961-969.
• Et M, Kandemir HŞ, Çakallı H. 2021. △^m-weighted statistical convergence. AIP Conf Proc, 2334: 040005.
• Fast H. 1951. Sur la convergence statistique. Colloquium Mathematicum, 2: 241-244.
• Fridy J. 1985. On statistical convergence, Analysis, 5: 301-313.
• Ghosal S. 2016. Weighted statistical convergence of order α and its applications. J Egyptian Math Soc, 24(1): 60-67.
• Güngör M, Et M. 2003. △^r-strongly almost summable sequences defined by Orlicz functions. Indian J Pure Appl Math, 34(8): 1141-1151.
• Kadak U. 2016. On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables. J Math Analy Appl, 443(2): 752-764.
• Kandemir HŞ, Et M, Çakallı H. 2023. Weighted statistical convergence of order α. Facta Univ Series: Math Info, 38(2): 317-327.
• Karakaya V, Chishti TA. 2009. Weighted statistical convergence. Iranian J Sci Technol Transact A: Sci, 33(33): 219-223.
• Kızmaz K. 1981. On certain sequence spaces. Canadian Math Bull, 24(2): 169-176.
• Mursaleen M, Karakaya V, Ertürk M, Gürsoy F. 2012. Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl Math Comput, 218(18): 9132-9137.
• Salat T. 1980. On statistically convergent sequences of real numbers. MathSlovaca, 30: 139-150.
• Schoenberg IJ. 959. The integrability of certain functions and related summability methods. Amer Math Monthly, 66: 361-375.
• Steinhaus H. 1951. Sur la convergence ordinaire et la convergence asymptotique. Colloquium Math, 2: 73-74.