Mahler's theorem

In mathematics, Mahler's theorem, introduced by {{harvs|txt|authorlink=Kurt Mahler|first=Kurt|last=Mahler|year=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 $(\Delta^n f)(0) = \sum^n_{k=0} (-1)^{n-k} \binom{n}{k} f(k).$ 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.

References

{{Citation | last1=Mahler | first1=K. | title=An interpolation series for continuous functions of a p-adic variable | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846 | mr=0095821 | year=1958 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=1958 | issue=199 | pages=23–34| doi=10.1515/crll.1958.199.23 | s2cid=199546556 | url-access=subscription }}

Category:Factorial and binomial topics

Category:Theorems in mathematical analysis