Mahler’s theorem for continuous functions on the $p$-adic integers

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 $\|f\|_{\infty}=\sup_{n\geq 0}|a_{n}|_{p}$, where $\|\cdot\|_{\infty}$ denotes the sup norm.

Title Mahler’s theorem for continuous functions on the $p$-adic integers MahlersTheoremForContinuousFunctionsOnThePadicIntegers 2013-03-22 18:32:07 2013-03-22 18:32:07 azdbacks4234 (14155) azdbacks4234 (14155) 6 azdbacks4234 (14155) Theorem msc 11S80