en>Cydebot: Robot - Moving category One to :Category:1 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.

2014-11-28T22:01:50Z

Robot - Moving category One to Category:1 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.

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	~~{{distinguish\|Mahler's compactness theorem}}~~		Hello! My name is Zelma. <br>It is a little about myself: I live in Canada, my city of London. <br>It's called often Northern or cultural capital of ON. I've married 1 years ago.<br>I have two children - a son (Sherrie) and the daughter (Opal). We all like Art collecting.<br><br>Take a look at my blog post ... make up online; [http://bestbreastfirmingcream.com just click the following internet page],
	~~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]~~]

en>Tamfang: add a necessary condition; remove something not distinctive of unit cubes; split a paragraph of two unrelated sentences

2013-12-22T21:51:01Z

add a necessary condition; remove something not distinctive of *unit* cubes; split a paragraph of two unrelated sentences

New page

{{distinguish|Mahler's compactness theorem}}
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]]

Unit cube - Revision history

en>Cydebot: Robot - Moving category One to :Category:1 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.

en>Tamfang: add a necessary condition; remove something not distinctive of *unit* cubes; split a paragraph of two unrelated sentences

en>Tamfang: add a necessary condition; remove something not distinctive of unit cubes; split a paragraph of two unrelated sentences