Mahler's 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 [math]\displaystyle{ (\Delta f)(x)=f(x+1)-f(x) }[/math] be the forward difference operator. Then for any p-adic function [math]\displaystyle{ f: \mathbb{Z}_p \to \mathbb{Q}_p }[/math], Mahler's theorem states that [math]\displaystyle{ f }[/math] is continuous if and only if its Newton series converges everywhere to [math]\displaystyle{ f }[/math], so that for all [math]\displaystyle{ x \in \mathbb{Z}_p }[/math] we have
- [math]\displaystyle{ f(x)=\sum_{n=0}^\infty (\Delta^n f)(0){x \choose n}, }[/math]
where
- [math]\displaystyle{ {x \choose n}=\frac{x(x-1)(x-2)\cdots(x-n+1)}{n!} }[/math]
is the [math]\displaystyle{ n }[/math]th binomial coefficient polynomial. Here, the [math]\displaystyle{ n }[/math]th forward difference is computed by the binomial transform, so that[math]\displaystyle{ (\Delta^n f)(0) = \sum^n_{k=0} (-1)^{n-k} \binom{n}{k} f(k). }[/math]Moreover, we have that [math]\displaystyle{ f }[/math] is continuous if and only if the coefficients [math]\displaystyle{ (\Delta^n f)(0) \to 0 }[/math] in [math]\displaystyle{ \mathbb{Q}_p }[/math] as [math]\displaystyle{ n \to \infty }[/math].
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, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846
 |  |
