mahler's theorem

In mathematics, Mahler's theorem, introduced by , 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 $f: \mathbb_p \to \mathbb_p$, Mahler's theorem states that $f$ is continuous if and only if its Newton series converges everywhere to $f$, so that for all $x \in \mathbb_p$ we have

:$f(x)=\sum_^\infty (\Delta^n f)(0),$

where

:$=\frac$

is the $n$th binomial coefficient polynomial. Here, the $n$th forward difference is computed by the binomial transform, so that$(\Delta^n f)(0) = \sum^n_ (-1)^ \binom f(k).$Moreover, we have that $f$ is continuous if and only if the coefficients $(\Delta^n f)(0) \to 0$ in $\mathbb_p$ 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

Factorial and binomial topics
Theorems in mathematical analysis