= Mahler expansion of continuous p-adic functions
{c}
{title2=$f(x)=\sum_{n\geq0}a_n\binom{x}{n},\quad a_n\to0$}
Let $I$ be a complete <discrete valuation ring> with a continuous $\mathbb Z_p$-algebra structure. Every <continuous function> on $\mathbb Z_p$ with values in $I$ has a unique uniformly convergent expansion in the <binomial polynomials>, with $a_n=\Delta^nf(0)$. This is the coefficient-ring version of the <Mahler theorem>. To see that $a_n\to0$, approximate the function modulo any power of the <maximal ideal> by a function of period $p^k$. In the expansion of $(E-1)^{p^k}$, all intermediate <binomial coefficients> are divisible by $p$, while $(E^{p^k}-1)$ kills the periodic function. Iterating makes its differences vanish modulo the prescribed ideal power. <Finite differences> do not increase the <uniform norm>, so the same conclusion holds for the original function. <Binomial inversion> gives agreement at the nonnegative <integers>, and their density in the <p-adic integers> gives the expansion everywhere. This gives the dual description of <p-adic measures> by arbitrary bounded sequences of binomial moments, and hence the <Amice transform>.
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