Mahler's theorem
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In mathematics, Mahler's theorem, named after Kurt Mahler (1903–1988), identifies one of various respects in which analysis is simpler with p-adic numbers than with real numbers.
In any field, one has the following result. Let
- (Δf)(x) = f(x + 1) − f(x)
be the forward difference operator. Then for polynomial functions f we have the Newton series:
where
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 complex number field are far more tightly constrained, and require Carlson's theorem to hold.
It is a fact of algebra that if f is a polynomial function with coefficients in any field of characteristic 0, the same identity holds where the sum has finitely many terms.
