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 ${\mathrm{Q}}_p$, and for each $n\mathrm{\in }\mathrm{N}$, put $a_n\mathrm{=}{\mathrm{\sum }}_{i\mathrm{=}\mathrm{0}}^n{\mathrm{(}\mathrm{-}\mathrm{1}\mathrm{)}}^{n\mathrm{-}i}\mathit{}\mathrm{\left(}\genfrac{}{}{0pt}{}{n}{i}\mathrm{\right)}\mathit{}f\mathit{}\mathrm{(}i\mathrm{)}$. Then $a_n\mathrm{\to }\mathrm{0}$ as $n\mathrm{\to }\mathrm{\infty }$, the series ${\mathrm{\sum }}_{n\mathrm{=}\mathrm{0}}^{\mathrm{\infty }}{a}_n\mathit{}\mathrm{\left(}\genfrac{}{}{0pt}{}{\mathrm{\cdot }}{n}\mathrm{\right)}$ converges uniformly to $f$ on ${\mathrm{Z}}_p$, and ${\mathrm{\parallel }f\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{=}{\sup }_{n\mathrm{\ge }\mathrm{0}}\mathit{}{\mathrm{|}{a}_n\mathrm{|}}_p$, where $\mathrm{\parallel }\mathrm{\cdot }{\mathrm{\parallel }}_{\mathrm{\infty }}$ denotes the sup norm.

Title	Mahler’s theorem for continuous functions on the $p$-adic integers
Canonical name	MahlersTheoremForContinuousFunctionsOnThePadicIntegers
Date of creation	2013-03-22 18:32:07
Last modified on	2013-03-22 18:32:07
Owner	azdbacks4234 (14155)
Last modified by	azdbacks4234 (14155)
Numerical id	6
Author	azdbacks4234 (14155)
Entry type	Theorem
Classification	msc 11S80