= Mahler theorem
{c}
{title2=$f(x)=\sum_{n\geq0}a_n\binom xn$}
{wiki=Mahler's_theorem}
= Mahler expansion
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{synonym}
Every function in <continuous functions on the p-adic integers> has a unique uniformly convergent expansion in the <binomial polynomials>, with $a_n\to0$ and $a_n=(\Delta^nf)(0)$. Conversely, any coefficient sequence tending to zero defines such a <continuous function>. To prove the expansion once $a_n\to0$ is known, use $|\binom xn|_p\leq1$ for uniform convergence. Finite binomial inversion gives agreement with $f$ at each nonnegative integer, and density gives agreement everywhere. The coefficients recover successively from these integer values, proving uniqueness. Moreover $\|f\|_\infty=\sup_n|a_n|_p$, because finite differences bound each coefficient by the norm and the expansion gives the reverse inequality.
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