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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Yaroslavl State University
2016-02-01
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| Series: | Modelirovanie i Analiz Informacionnyh Sistem |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/302 |
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doaj-d099007aa89a4831b27dc88fb853bb442021-07-29T08:15:21ZengYaroslavl State UniversityModelirovanie i Analiz Informacionnyh Sistem1818-10152313-54172016-02-0123151110.18255/1818-1015-2016-1-5-11278Asymptotic 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 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
E. A. Timofeev |
| spellingShingle |
E. A. Timofeev Asymptotic Formula for the Moments of Takagi Function Modelirovanie i Analiz Informacionnyh Sistem moments self-similar takagi function singular mellin transform asymptotic |
| author_facet |
E. A. Timofeev |
| author_sort |
E. A. Timofeev |
| title |
Asymptotic Formula for the Moments of Takagi Function |
| title_short |
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_sort |
asymptotic formula for the moments of takagi function |
| publisher |
Yaroslavl State University |
| series |
Modelirovanie i Analiz Informacionnyh Sistem |
| issn |
1818-1015 2313-5417 |
| publishDate |
2016-02-01 |
| 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 . |
| 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 |
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1721256512805928960 |