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.

单词	MahlersTheoremForContinuousFunctionsOnThePadicIntegers
释义	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.
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Mahler’s theorem for continuous functions on the p-adic integers

Theorem.

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