Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916729432571904 |
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| author | Bhat, Aaqib Ayoub Khan, Asif Mursaleen, M. |
| author_facet | Bhat, Aaqib Ayoub Khan, Asif Mursaleen, M. |
| contents | This paper presents a new approach of constructing $α$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $α$-fractals utilizing neural network-based operators, providing an approach to generating fractal functions with interpolation properties. Based on the same foundation, we have developed fractal interpolation functions that utilize only the values of the original function at the nodes or partition points, unlike traditional methods that rely on the entire original function.
Further, we have constructed \(α\)-fractals that preserve the smoothness of functions under certain constraints by employing a four-layered neural network operator, ensuring that if \(f \in C^{r}[a,b]\), then the corresponding fractal \(f^α \in C^{r}[a,b]\). Furthermore, we analyze the convergence of these $α$-fractals to the original function under suitable conditions. The work uses key approximation theory tools, such as the modulus of continuity and interpolation operators, to develop convergence results and uniform approximation error bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis Bhat, Aaqib Ayoub Khan, Asif Mursaleen, M. Machine Learning Numerical Analysis 28A80, 41A05, 41A25, 41A29, 41A30, 65D05 This paper presents a new approach of constructing $α$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $α$-fractals utilizing neural network-based operators, providing an approach to generating fractal functions with interpolation properties. Based on the same foundation, we have developed fractal interpolation functions that utilize only the values of the original function at the nodes or partition points, unlike traditional methods that rely on the entire original function. Further, we have constructed \(α\)-fractals that preserve the smoothness of functions under certain constraints by employing a four-layered neural network operator, ensuring that if \(f \in C^{r}[a,b]\), then the corresponding fractal \(f^α \in C^{r}[a,b]\). Furthermore, we analyze the convergence of these $α$-fractals to the original function under suitable conditions. The work uses key approximation theory tools, such as the modulus of continuity and interpolation operators, to develop convergence results and uniform approximation error bounds. |
| title | Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis |
| topic | Machine Learning Numerical Analysis 28A80, 41A05, 41A25, 41A29, 41A30, 65D05 |
| url | https://arxiv.org/abs/2505.06229 |