The higher order partial derivatives of Okamoto's function with respect to the parameter
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| Format: | Preprint |
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2025
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| _version_ | 1866911018699980800 |
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| author | Allaart, Pieter Dalaklis, Nathan Kawamura, Kiko Ortiz, Matthew Zheng, Jiajie |
| author_facet | Allaart, Pieter Dalaklis, Nathan Kawamura, Kiko Ortiz, Matthew Zheng, Jiajie |
| contents | Let $\{F_a: a\in(0,1)\}$ be Okamoto's family of continuous self-affine functions, introduced in [{\em Proc. Japan Acad. Ser. A Math. Sci.} {\bf 81} (2005), no. 3, 47--50]. This family includes well-known ``pathological" examples such as Cantor's devil's staircase and Perkins' continuous but nowhere differentiable function. It is well known that $F_a(x)$ is real analytic in $a$ for every $x\in[0,1]$. We introduce the functions \[ M_{k,a}(x):=\frac{\partial^k}{\partial a^k}F_a(x), \qquad k\in\mathbb{N}, \quad x\in[0,1]. \] We compute the box-counting dimension of the graph of $M_{k,a}$, characterize its differentiability, and investigate in detail the set of points where $M_{k,a}$ has an infinite derivative. While some of our results are similar to the known facts about Okamoto's function, there are also some notable differences and surprising new phenomena that arise when considering the higher order partial derivatives of $F_a$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17737 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The higher order partial derivatives of Okamoto's function with respect to the parameter Allaart, Pieter Dalaklis, Nathan Kawamura, Kiko Ortiz, Matthew Zheng, Jiajie Classical Analysis and ODEs Primary: 26A27, 26A30, Secondary: 28A78, 11A63 Let $\{F_a: a\in(0,1)\}$ be Okamoto's family of continuous self-affine functions, introduced in [{\em Proc. Japan Acad. Ser. A Math. Sci.} {\bf 81} (2005), no. 3, 47--50]. This family includes well-known ``pathological" examples such as Cantor's devil's staircase and Perkins' continuous but nowhere differentiable function. It is well known that $F_a(x)$ is real analytic in $a$ for every $x\in[0,1]$. We introduce the functions \[ M_{k,a}(x):=\frac{\partial^k}{\partial a^k}F_a(x), \qquad k\in\mathbb{N}, \quad x\in[0,1]. \] We compute the box-counting dimension of the graph of $M_{k,a}$, characterize its differentiability, and investigate in detail the set of points where $M_{k,a}$ has an infinite derivative. While some of our results are similar to the known facts about Okamoto's function, there are also some notable differences and surprising new phenomena that arise when considering the higher order partial derivatives of $F_a$. |
| title | The higher order partial derivatives of Okamoto's function with respect to the parameter |
| topic | Classical Analysis and ODEs Primary: 26A27, 26A30, Secondary: 28A78, 11A63 |
| url | https://arxiv.org/abs/2506.17737 |