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Mahler's theorem for continuous functions on the  -adic integers
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(Theorem)
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Theorem (Mahler) Let $f$ be a continuous function on the $p$ -adic integers taking values in some finite extension $K$ of $\mathbb{Q}_p$ , and for each $n\in\mathbb{N}$ , put $a_n=\sum_{i=0}^n(-1)^{n-i}\tbinom{n}{i}f(i)$ . Then $a_n\rightarrow 0$ as $n\rightarrow\infty$ , the series $\sum_{n=0}^\infty a_n\tbinom{\cdot}{n}$ converges uniformly to $f$ on $\mathbb{Z}_p$ , and $\norm{f}_\infty=\sup_{n\geq 0}\abs{a_n}_p$ , where $\norm{\cdot}_\infty$ denotes the sup norm.
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"Mahler's theorem for continuous functions on the  -adic integers" is owned by azdbacks4234.
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| Keywords: |
p-adic integers, p-adic analysis |
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Cross-references: sup norm, converges uniformly, series, finite extension, integers, continuous function
This is version 3 of Mahler's theorem for continuous functions on the  -adic integers, born on 2008-11-07, modified 2008-11-07.
Object id is 11243, canonical name is MahlersTheorem.
Accessed 465 times total.
Classification:
| AMS MSC: | 11S80 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Other analytic theory ) |
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Pending Errata and Addenda
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