Unit cube: Difference between revisions

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en>Tamfang
add a necessary condition; remove something not distinctive of *unit* cubes; split a paragraph of two unrelated sentences
 
en>Cydebot
 
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{{distinguish|Mahler's compactness theorem}}
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In mathematics, '''Mahler's theorem''', introduced by {{harvs|txt|authorlink=Kurt Mahler|first=Kurt|last=Mahler|year=1958}}, expresses continuous ''p''-adic functions in terms of polynomials.
 
In any [[field (mathematics)|field]], 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's theorem states that if ''f'' is a continuous [[p-adic number|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 ''k''th term is ''x''<sup>''k''</sup>.
 
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 (mathematics)|field]] of [[characteristic (algebra)|characteristic]] 0, the same identity holds where the sum has finitely many terms.
 
==References==
 
*{{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=199 | pages=23–34}}
 
[[Category:Factorial and binomial topics]]
[[Category:Theorems in analysis]]

Latest revision as of 23:01, 28 November 2014

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