Mahler's theorem

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

$f(x)=\sum _{n=0}^{\mathrm{\infty }}({\mathrm{\Delta }}^{n}f)(0)(\genfrac{}{}{0ex}{}{x}{n}),$

where

$(\genfrac{}{}{0ex}{}{x}{n})=\frac{x(x-1)(x-2)\cdots (x-n+1)}{n!}$

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

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.