On the $L^1$ and pointwise divergence of continuous functions
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| Format: | Preprint |
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2020
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| _version_ | 1866909153177370624 |
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| author | Gryszka, Karol Pasteczka, Paweł |
| author_facet | Gryszka, Karol Pasteczka, Paweł |
| contents | For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set
$$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$
The origin of this problem is the limit behaviour of quasiarithmetic means. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_12752 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the $L^1$ and pointwise divergence of continuous functions Gryszka, Karol Pasteczka, Paweł Classical Analysis and ODEs 26E60, 54C30, 26A15, 26A30, 28A78 For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means. |
| title | On the $L^1$ and pointwise divergence of continuous functions |
| topic | Classical Analysis and ODEs 26E60, 54C30, 26A15, 26A30, 28A78 |
| url | https://arxiv.org/abs/2007.12752 |