Mahler's theorem

Mahler's theorem Not to be confused with Mahler's compactness theorem.

In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses continuous p-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result: Let {displaystyle (Delta f)(x)=f(x+1)-f(x)} be the forward difference operator. Then for polynomial functions f we have the Newton series {displaystyle f(x)=sum _{k=0}^{infty }(Delta ^{k}f)(0){x choose k},} where {displaystyle {x choose k}={frac {x(x-1)(x-2)cdots (x-k+1)}{k!}}} is the kth binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function f is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity. Mahler's theorem states that if f is a continuous p-adic-valued function on the p-adic integers then the same identity holds. The relationship between the operator Δ and this polynomial sequence is much like that between differentiation and the sequence whose kth term is xk.

It is remarkable that as weak an assumption as continuity is enough; 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, 199: 23–34, ISSN 0075-4102, MR 0095821 Categories: Factorial and binomial topicsTheorems in analysis