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.
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
th binomial coefficient polynomial.
th forward difference is computed by the binomial transform, so that
{\displaystyle (\Delta ^{n}f)(0)=\sum _{k=0}^{n}(-1)^{n-k}{\binom {n}{k}}f(k).}
is continuous if and only if the coefficients
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.