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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.
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| In any [[field (mathematics)|field]], one has the following result. Let
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| :<math>(\Delta f)(x)=f(x+1)-f(x)\,</math>
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| be the forward [[difference operator]]. Then for [[polynomial function]]s ''f'' we have the [[Newton series]]:
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| :<math>f(x)=\sum_{k=0}^\infty (\Delta^k f)(0){x \choose k},</math>
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| where
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| :<math>{x \choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}</math>
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| is the ''k''th binomial coefficient polynomial.
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| 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]].
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| 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.
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| 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>.
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| 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.
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| 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.
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| ==References==
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| *{{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}}
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| [[Category:Factorial and binomial topics]]
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| [[Category:Theorems in analysis]]
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Hello! My name is Zelma.
It is a little about myself: I live in Canada, my city of London.
It's called often Northern or cultural capital of ON. I've married 1 years ago.
I have two children - a son (Sherrie) and the daughter (Opal). We all like Art collecting.
Take a look at my blog post ... make up online; just click the following internet page,