Mahler's theorem

Not to be confused with Mahler's compactness theorem.

In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. This series is referred to as a Mahler series or Mahler expansion, or as Mahler's expansion. 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 $$f(x):=\sum _{n=0}^{\infty }a_{n}{\binom {x}{n}}$$ 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

• Mahler, K. (1958), "An interpolation series for continuous functions of a p-adic variable", Journal für die reine und angewandte Mathematik, 1958 (199): 23–34, doi:10.1515/crll.1958.199.23, ISSN 0075-4102, MR 0095821, S2CID 199546556