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  be the forward difference operator. Then for any p-adic function , Mahler's theorem states that  is continuous if and only if its Newton series converges everywhere to , so that for all  we have

where

is the th binomial coefficient polynomial. Here, the th forward difference is computed by the binomial transform, so thatMoreover, we have that  is continuous if and only if the coefficients  in  as .

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

Factorial and binomial topics
Theorems in analysis