In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. This series is referred to as a Mahler series or Mahler expansion, or as Mahler's expansion. It is the p-adic counterpart to the Stone–Weierstrass theorem for continuous real-valued functions on a closed interval.
Statement Let ( Δ f ) ( x ) = f ( x + 1 ) − f ( x ) {\displaystyle (\Delta f)(x)=f(x+1)-f(x)} be the forward difference operator. Then for any p-adic function f : Z p → Q p {\displaystyle f:\mathbb {Z} _{p}\to \mathbb {Q} _{p}} , Mahler's theorem states that f {\displaystyle f} is continuous if and only if its Newton series converges everywhere to f {\displaystyle f} , so that for all x ∈ Z p {\displaystyle x\in \mathbb {Z} _{p}} we have
f ( x ) = ∑ n = 0 ∞ ( Δ n f ) ( 0 ) ( x n ) , {\displaystyle f(x)=\sum _{n=0}^{\infty }(\Delta ^{n}f)(0){x \choose n},}
where
( x n ) = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) n ! {\displaystyle {x \choose n}={\frac {x(x-1)(x-2)\cdots (x-n+1)}{n!}}}
is the n {\displaystyle n} th binomial coefficient polynomial. Here, the n {\displaystyle n} th forward difference is computed by the binomial transform, so that ( Δ n f ) ( 0 ) = ∑ k = 0 n ( − 1 ) n − k ( n k ) f ( k ) . {\displaystyle (\Delta ^{n}f)(0)=\sum _{k=0}^{n}(-1)^{n-k}{\binom {n}{k}}f(k).} Moreover, we have that f ( x ) := ∑ n = 0 ∞ a n ( x n ) {\displaystyle f(x):=\sum _{n=0}^{\infty }a_{n}{\binom {x}{n}}} is continuous if and only if the coefficients a n = ( Δ n f ) ( 0 ) → 0 {\displaystyle a_{n}=(\Delta ^{n}f)(0)\to 0} in Q p {\displaystyle \mathbb {Q} _{p}} as n → ∞ {\displaystyle 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
Mahler, K. (1958), "An interpolation series for continuous functions of a p-adic variable", Journal für die reine und angewandte Mathematik, 1958 (199): 23–34, doi:10.1515/crll.1958.199.23, ISSN 0075-4102, MR 0095821, S2CID 199546556