A new class of positive linear operators preserving logarithmic functions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915855963521024 |
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| author | Angeloni, Laura Costarelli, Danilo Darielli, Chiara |
| author_facet | Angeloni, Laura Costarelli, Danilo Darielli, Chiara |
| contents | In this paper, we introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that reproduce the logarithmic function $\ln(1+μ+x)$, with $μ> 0$ and $x \in [0,1]$. We prove pointwise and uniform convergence and we derive a quantitative estimate of the approximation error in terms of the modulus of continuity. We also obtain a Voronovskaja-type asymptotic formula, that is used to establish saturation results and inverse theorems. In particular, the saturation class of the considered approximation process is characterized by solving a second order differential equation. Shape-preserving properties, such as monotonicity, concavity and variation diminishing, are also investigated. Finally, a simple application to signal denoising is addressed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06168 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new class of positive linear operators preserving logarithmic functions Angeloni, Laura Costarelli, Danilo Darielli, Chiara Functional Analysis In this paper, we introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that reproduce the logarithmic function $\ln(1+μ+x)$, with $μ> 0$ and $x \in [0,1]$. We prove pointwise and uniform convergence and we derive a quantitative estimate of the approximation error in terms of the modulus of continuity. We also obtain a Voronovskaja-type asymptotic formula, that is used to establish saturation results and inverse theorems. In particular, the saturation class of the considered approximation process is characterized by solving a second order differential equation. Shape-preserving properties, such as monotonicity, concavity and variation diminishing, are also investigated. Finally, a simple application to signal denoising is addressed. |
| title | A new class of positive linear operators preserving logarithmic functions |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2602.06168 |