Asymptotic Formula for the Moments of Takagi Function
Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by T(x) = ∞ ᢘ k=0 2−nρ(2nx), where ρ(x) = min k∈Z |x − k|. The moments of Takagi function are defined as Mn = ᝈ 1 0 xnT(x) dx. The main result of this paper is the following: Mn = lnn − Γ(1) − lnπ n...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2016-02-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/302 |
| _version_ | 1826558991457583104 |
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| author | E. A. Timofeev |
| author_facet | E. A. Timofeev |
| author_sort | E. A. Timofeev |
| collection | DOAJ |
| description | Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by T(x) = ∞ ᢘ k=0 2−nρ(2nx), where ρ(x) = min k∈Z |x − k|. The moments of Takagi function are defined as Mn = ᝈ 1 0 xnT(x) dx. The main result of this paper is the following: Mn = lnn − Γ(1) − lnπ n2 ln 2 + 1 2n2 + 2 n2 ln 2 φ(n) + O(n−2.99), where φ(x) = ᝨ kᡘ=0 Γ ᝈ2πik ln 2 ζ ᡸ2πik ln 2 ᡸ x−2lπni2k . |
| first_indexed | 2024-04-10T02:24:20Z |
| format | Article |
| id | doaj.art-d099007aa89a4831b27dc88fb853bb44 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2025-03-14T08:53:17Z |
| publishDate | 2016-02-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-d099007aa89a4831b27dc88fb853bb442025-03-02T12:46:57ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-02-0123151110.18255/1818-1015-2016-1-5-11278Asymptotic Formula for the Moments of Takagi FunctionE. A. Timofeev0Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, RussiaTakagi function is a simple example of a continuous but nowhere differentiable function. It is defined by T(x) = ∞ ᢘ k=0 2−nρ(2nx), where ρ(x) = min k∈Z |x − k|. The moments of Takagi function are defined as Mn = ᝈ 1 0 xnT(x) dx. The main result of this paper is the following: Mn = lnn − Γ(1) − lnπ n2 ln 2 + 1 2n2 + 2 n2 ln 2 φ(n) + O(n−2.99), where φ(x) = ᝨ kᡘ=0 Γ ᝈ2πik ln 2 ζ ᡸ2πik ln 2 ᡸ x−2lπni2k .https://www.mais-journal.ru/jour/article/view/302momentsself-similartakagi functionsingularmellin transformasymptotic |
| spellingShingle | E. A. Timofeev Asymptotic Formula for the Moments of Takagi Function Моделирование и анализ информационных систем moments self-similar takagi function singular mellin transform asymptotic |
| title | Asymptotic Formula for the Moments of Takagi Function |
| title_full | Asymptotic Formula for the Moments of Takagi Function |
| title_fullStr | Asymptotic Formula for the Moments of Takagi Function |
| title_full_unstemmed | Asymptotic Formula for the Moments of Takagi Function |
| title_short | Asymptotic Formula for the Moments of Takagi Function |
| title_sort | asymptotic formula for the moments of takagi function |
| topic | moments self-similar takagi function singular mellin transform asymptotic |
| url | https://www.mais-journal.ru/jour/article/view/302 |
| work_keys_str_mv | AT eatimofeev asymptoticformulaforthemomentsoftakagifunction |