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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.02557 |
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| _version_ | 1866914017452228608 |
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| author | Söylemez, Dilek Ünver, Mehmet |
| author_facet | Söylemez, Dilek Ünver, Mehmet |
| contents | This paper establishes an abstract Korovkin-type approximation theorem in general spaces, extending the framework of approximation theory to accommodate broader contexts. A critical result supporting this theorem is the proof that any $P$-statistically convergent sequence contains a classically convergent subsequence over a density $1$ set, which plays a foundational role in the analysis. As a conclusion, we investigate the convergence of the $r$-th order generalization of linear operators, which may lack positivity, and present a Korovkin-type approximation theorem for periodic functions, both utilizing $P$-statistical convergence. These contributions generalize and improve existing results in approximation theory, providing novel insights and methodologies, supported by practical examples and corollaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02557 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power series statistical convergence and an abstract Korovkin-type approximation theorem Söylemez, Dilek Ünver, Mehmet Functional Analysis This paper establishes an abstract Korovkin-type approximation theorem in general spaces, extending the framework of approximation theory to accommodate broader contexts. A critical result supporting this theorem is the proof that any $P$-statistically convergent sequence contains a classically convergent subsequence over a density $1$ set, which plays a foundational role in the analysis. As a conclusion, we investigate the convergence of the $r$-th order generalization of linear operators, which may lack positivity, and present a Korovkin-type approximation theorem for periodic functions, both utilizing $P$-statistical convergence. These contributions generalize and improve existing results in approximation theory, providing novel insights and methodologies, supported by practical examples and corollaries. |
| title | Power series statistical convergence and an abstract Korovkin-type approximation theorem |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2509.02557 |