Mahler's theorem

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 nth binomial coefficient polynomial. Here, the nth 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(x):=\sum_{n=0}^\infty a_n\binom{x}{n} is continuous if and only if the coefficients a_n=(\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.

References