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In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), 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 $(\Delta ^{n}f)(0)=\sum _{k=0}^{n}(-1)^{n-k}{\binom {n}{k}}f(k).$ 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.