Mahler's theorem for continuous functions on the $p$-adic integers
MahlersTheorem
2008-11-07 18:20:31
2008-11-07 19:03:16
Theorem
p-adic integers
p-adic analysis
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\begin{thm*}
(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.
\end{thm*}