Papyrus 45 - Revision history

en>Magioladitis: WPCleaner v1.31 - Fixed using WP:WCW (Superscript tag with no correct end)

2014-04-23T15:16:06Z

WPCleaner v1.31 - Fixed using WP:WCW (Superscript tag with no correct end)

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	~~In mathematics, the~~ '~~''Stieltjes transformation''' ''S''<sub>ρ</sub>(''z'') of a measure of density ρ~~ on ~~a real interval ''I''~~ is the ~~function of the complex variable~~ '~~'z'' defined outside ''I'' by the formula~~		Golda is what's created on my birth certificate even though it is not the name on my birth certificate. Alaska is where he's usually been residing. Doing ballet is some thing she would never give up. For many years she's been operating as a journey agent.<br><br>Here is my homepage - free psychic, [http://Galab-work.cs.Pusan.ac.kr/Sol09B/?document_srl=1489804 just click the next document],

	~~:<math>S_{\rho}(z)=\int_I\frac{\rho(t)\,dt}{z-t}~~.~~</math>~~

	~~Under certain conditions we can reconstitute the density function ρ starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron~~. For ~~example, if the density ρ is continuous throughout~~ '~~'I'', one will have inside this interval~~

	~~:<math>\rho(x)=\underset{\varepsilon\rightarrow 0^+}{\text{lim}}\frac{S_{\rho}(x-i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}.</math>~~

	~~==Connections with moments of measures==~~
	~~{{Main\|moment problem}}~~

	~~If the measure of density ρ has [[moment (mathematics)\|moments]] of any order defined for each integer by the equality~~

	~~:<math>m_{n}=\int_I t^n\,\rho(t)\,dt,</math>~~

	~~then the [[Stieltjes]] transformation of ρ admits for each integer ''n'' the [[Asymptotic analysis\|asymptotic]] expansion in the neighbourhood of infinity given by~~

	~~:<math>S_{\rho}(z)=\sum_{k=0}^{k=n}\frac{m_k}{z^{k+1}}+o\left(\frac{1}{z^{n+1}}\right).</math>~~

	~~Under certain conditions the complete expansion~~ as a ~~[[Laurent series]] can be obtained:~~

	~~:<math>S_{\rho}(z)=\sum_{n=0}^{n=\infty}\frac{m_n}{z^{n+1}}.</math>~~

	~~==Relationships to orthogonal polynomials==~~

	~~The correspondence <math>(f,g)\mapsto \int_I f(t)g(t)\rho(t)\,dt</math> defines an [[inner product]] on the space of [[continuous function]]s on the interval ''I''~~.

	~~If {''P~~<~~sub~~>n<~~/sub~~>~~''}~~ is ~~a sequence of [[orthogonal polynomials]] for this product~~, ~~we can create the sequence of associated [~~[~~secondary polynomials]] by the formula~~

	: ~~<math>Q_n(x)=\int_I \frac{P_n (t)-P_n (x)}{t-x}\rho (t)\,dt.<~~/~~math>~~

	~~It appears that <math>F_n(z)=\frac{Q_n(z)}{P_n(z)}<~~/~~math> is a [[Padé approximation]] of ''S''<sub>ρ</sub>(''z'') in a neighbourhood of infinity, in the sense that~~

	~~: <math>S_\rho(z)~~-~~\frac{Q_n(z)}{P_n(z)}=O\left(\frac{1}{z^{2n}}\right)~~.~~</math>~~

	Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a [[generalized continued fraction\|continued fraction]] for the Stieltjes transformation whose successive [[Convergent (continued fraction)\|convergents]] are the fractions ''F<sub>n</sub>''(''z'').

	~~The Stieltjes transformation can also be used to construct from the density ρ an effective measure for transforming the secondary polynomials into an orthogonal system~~. ~~(For more details see the article [[secondary measure]]~~.)

	=~~=See also==~~

	* [[Orthogonal polynomials]]
	* [[Secondary polynomials]]
	* [[Secondary measure]]

	~~==References==~~
	*{{cite book\|author = H. S. Wall\|title = Analytic Theory of Continued Fractions\|publisher = D. Van Nostrand Company Inc.\|year = 1948}}

	~~[[Category:Integral transforms]]~~
	~~[[Category:Continued fractions]~~]

en>Das Kleinhirn: Change URL

2013-06-10T18:56:33Z

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← Older revision		Revision as of 20:56, 10 June 2013
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	I ~~would like to introduce myself to you,~~ I ~~am Andrew and my spouse doesn~~'t ~~like it at all~~. ~~Kentucky~~ is ~~exactly where~~ I'~~ve usually been residing~~. ~~The preferred hobby~~ for ~~him and his kids is to play lacross and he~~'ll be ~~beginning something else alongside with it~~. I ~~am currently a travel agent~~.<br><br>~~Have~~ a ~~look at my web~~-~~site :: spirit messages~~ ([~~http~~://1.~~234~~.36.~~240/fxac/m001_2/7330 visit my webpage~~])		In mathematics, the '''Stieltjes transformation''' ''S''<sub>ρ</sub>(''z'') of a measure of density ρ on a real interval ''I'' is the function of the complex variable ''z'' defined outside ''I'' by the formula

			:<math>S_{\rho}(z)=\int_I\frac{\rho(t)\,dt}{z-t}.</math>

			Under certain conditions we can reconstitute the density function ρ starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density ρ is continuous throughout ''I'', one will have inside this interval

			:<math>\rho(x)=\underset{\varepsilon\rightarrow 0^+}{\text{lim}}\frac{S_{\rho}(x-i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}.</math>

			==Connections with moments of measures==
			{{Main\|moment problem}}

			If the measure of density ρ has [[moment (mathematics)\|moments]] of any order defined for each integer by the equality

			:<math>m_{n}=\int_I t^n\,\rho(t)\,dt,</math>

			then the [[Stieltjes]] transformation of ρ admits for each integer ''n'' the [[Asymptotic analysis\|asymptotic]] expansion in the neighbourhood of infinity given by

			:<math>S_{\rho}(z)=\sum_{k=0}^{k=n}\frac{m_k}{z^{k+1}}+o\left(\frac{1}{z^{n+1}}\right).</math>

			Under certain conditions the complete expansion as a [[Laurent series]] can be obtained:

			:<math>S_{\rho}(z)=\sum_{n=0}^{n=\infty}\frac{m_n}{z^{n+1}}.</math>

			==Relationships to orthogonal polynomials==

			The correspondence <math>(f,g)\mapsto \int_I f(t)g(t)\rho(t)\,dt</math> defines an [[inner product]] on the space of [[continuous function]]s on the interval ''I''.

			If {''P<sub>n</sub>''} is a sequence of [[orthogonal polynomials]] for this product, we can create the sequence of associated [[secondary polynomials]] by the formula

			: <math>Q_n(x)=\int_I \frac{P_n (t)-P_n (x)}{t-x}\rho (t)\,dt.</math>

			It appears that <math>F_n(z)=\frac{Q_n(z)}{P_n(z)}</math> is a [[Padé approximation]] of ''S''<sub>ρ</sub>(''z'') in a neighbourhood of infinity, in the sense that

			: <math>S_\rho(z)-\frac{Q_n(z)}{P_n(z)}=O\left(\frac{1}{z^{2n}}\right).</math>

			Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a [[generalized continued fraction\|continued fraction]] for the Stieltjes transformation whose successive [[Convergent (continued fraction)\|convergents]] are the fractions ''F<sub>n</sub>''(''z'').

			The Stieltjes transformation can also be used to construct from the density ρ an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article [[secondary measure]].)

			==See also==

			* [[Orthogonal polynomials]]
			* [[Secondary polynomials]]
			* [[Secondary measure]]

			==References==
			*{{cite book\|author = H. S. Wall\|title = Analytic Theory of Continued Fractions\|publisher = D. Van Nostrand Company Inc.\|year = 1948}}

			[[Category:Integral transforms]]
			[[Category:Continued fractions]]

en>AbimaelLevid: es:Papiro 45

2011-12-12T20:15:31Z

es:Papiro 45

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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Kentucky is exactly where I've usually been residing. The preferred hobby for him and his kids is to play lacross and he'll be beginning something else alongside with it. I am currently a travel agent.<br><br>Have a look at my web-site :: spirit messages ([http://1.234.36.240/fxac/m001_2/7330 visit my webpage])