Mahler's theorem

In mathematics, Mahler's theorem, introduced by, expresses any continuous p-adic function as an infinite series of certain special polynomials. It is the p-adic counterpart to the Stone-Weierstrass theorem for continuous real-valued functions on a closed interval.

Statement
Let $$(\Delta f)(x)=f(x+1)-f(x)$$ be the forward difference operator. Then for any p-adic function $$f: \mathbb{Z}_p \to \mathbb{Q}_p$$, Mahler's theorem states that $$f$$ is continuous if and only if its Newton series converges everywhere to $$f$$, so that for all $$x \in \mathbb{Z}_p$$ we have


 * $$f(x)=\sum_{n=0}^\infty (\Delta^n f)(0){x \choose n},$$

where


 * $${x \choose n}=\frac{x(x-1)(x-2)\cdots(x-n+1)}{n!}$$

is the $$n$$th binomial coefficient polynomial. Here, the $$n$$th forward difference is computed by the binomial transform, so that$$ (\Delta^n f)(0) = \sum^n_{k=0} (-1)^{n-k} \binom{n}{k} f(k).$$Moreover, we have that $$f$$ is continuous if and only if the coefficients $$(\Delta^n f)(0) \to 0$$ in $$\mathbb{Q}_p$$ as $$n \to \infty$$.

It is remarkable that as weak an assumption as continuity is enough in the p-adic setting to establish convergence of Newton series. By contrast, Newton series on the field of complex numbers are far more tightly constrained, and require Carlson's theorem to hold.