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 f is continuous if and only if its Newton series converges everywhere to f, so that for all we have where is the nth binomial coefficient polynomial. Here, the nth forward difference is computed by the binomial transform, so thatMoreover, we have that f 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.