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. It is the p-adic counterpart to the Stone-Weierstrass theorem for continuous real-valued functions on a closed interval.

Statement

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

${\displaystyle f(x)=\sum _{n=0}^{\infty }(\Delta ^{n}f)(0){x \choose n},}$

where

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

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

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