Mahler's theorem

In mathematics, Mahler's theorem, introduced by , expresses continuous ''p''-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result:

Let $(\Delta f)(x)=f(x+1)-f(x)$ be the forward difference operator. Then for polynomial functions ''f'' we have the Newton series

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

where

:$=\frac$

is the ''k''th binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function ''f'' is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity. Mahler's theorem states that if ''f'' is a continuous p-adic-valued function on the ''p''-adic integers then the same identity holds. The relationship between the operator Δ and this polynomial sequence is much like that between differentiation and the sequence whose ''k''th term is ''x''^''k''.

It is remarkable that as weak an assumption as continuity is enough; 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=199 , pages=23–34

Factorial and binomial topics
Theorems in analysis