Mahler's theorem

Template:Short description Template:Distinguish In mathematics, Mahler's theorem, introduced by Template:Harvs, 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 $(\mathrm{\Delta }f)(x)=f(x+1)-f(x)$ 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

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

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 $$f(x):=\sum _{n=0}^{\mathrm{\infty }}{a}_n(\genfrac{}{}{0ex}{}{x}{n})$$ is continuous if and only if the coefficients $a_n=({\mathrm{\Delta }}^{n}f)(0)\to 0$ in ${\mathbb{Q}}_p$ 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.

References

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