Editing P-adic analysis (section)

===Mahler's theorem===
{{main article|Mahler's theorem}}
'''Mahler's theorem''', introduced by [[Kurt Mahler]],<ref>{{Citation | last1=Mahler | first1=K. | title=An interpolation series for continuous functions of a p-adic variable | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846 | mr=0095821 | year=1958 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1958 | issue=199 | pages=23–34| doi=10.1515/crll.1958.199.23 | s2cid=199546556 }}
</ref> expresses continuous ''p''-adic functions in terms of polynomials.

In any [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] 0, one has the following result.  Let

:<math>(\Delta f)(x)=f(x+1)-f(x)</math>

be the forward [[difference operator]].  Then for [[polynomial function]]s ''f'' we have the [[Newton series]]:

:<math>f(x)=\sum_{k=0}^\infty (\Delta^k f)(0){x \choose k},</math>

where

:<math>{x \choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}</math>

is the ''k''th 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 [[continuous function|continuity]].

Mahler proved the following result:

'''Mahler's theorem''': If ''f'' is a continuous [[p-adic number|''p''-adic]]-valued function on the ''p''-adic integers then the same identity holds.