Mahler's theorem

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.

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 {\displaystyle f} is continuous if and only if the coefficients ( Δ n f ) ( 0 ) → 0 {\displaystyle (\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