Directional statistics - Revision history

en>Aetheling: /* Distribution of the mean */ Fix another bad formula

2014-12-11T16:33:00Z

Distribution of the mean: Fix another bad formula

← Older revision		Revision as of 18:33, 11 December 2014
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	~~In [[mathematics]], especially [[functional analysis]], '''Bessel's inequality''' is a statement about the coefficients of an element <math>x</math> in a [[Hilbert space]] with respect~~ to ~~an [[orthonormal]] [[sequence]].~~		Nice to meet you, I am Marvella Shryock. California is our beginning location. What I love performing is to gather badges but I've been taking on new things lately. For many years he's been operating as a receptionist.<br><br>Feel free to surf to my web blog - meal delivery service ([http://www.Myprgenie.com/view-publication/weight-loss-getting-you-down-these-tips-can-help click through the next web page])

	~~Let <math>H</math> be a Hilbert space, and suppose that <math>e_1, e_2~~, ..~~.</math>~~ is ~~an orthonormal sequence in <math>H</math>~~. ~~Then, for any <math>x</math> in <math>H</math> one has~~
	~~:<math>\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2 </math>~~

	~~where 〈•,•〉 denotes the [[inner product space\|inner product]] in the Hilbert space <math>H</math>. If we define the infinite sum~~
	~~:<math>x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, </math>~~
	~~consisting of 'infinite sum' of [[vector resolute]] <math>x</math> in direction <math>e_k</math>, Bessel~~'s ~~[[inequality (mathematics)\|inequality]] tells us that this [[series (mathematics)\|series]] [[Limit of~~ a ~~sequence\|converges]]~~. ~~One can think of it that there exists~~ <~~math~~>~~x' \in H~~<~~/math~~> ~~which can be described in terms of potential basis <math>e_1, e_2, ...</math>.~~

	~~For a complete orthonormal sequence~~ (that is, for an orthonormal sequence which is a [[Orthonormal basis\|basis]]), we have [[Parseval's identity]], which replaces the inequality with an equality (and consequently <math> x'</math> with <math> x</math>).

	~~Bessel's inequality follows from the identity:~~
	:<math>0 \le \left\\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\\|^2 = \\|x\\|^2 - 2 \sum_{k=1}^n \|\langle x, e_k \rangle \|^2 + \sum_{k=1}^n \| \langle x, e_k \rangle \|^2 = \\|x\\|^2 - \sum_{k=1}^n \| \langle x, e_k \rangle \|^2,</math>
	~~which holds for any natural ''n''.~~

	~~==See also==~~
	* [[Cauchy–Schwarz inequality]]

	~~==External links==~~
	* {{springer\|title=Bessel inequality\|id=p/b015850}}
	* [http://~~mathworld~~.~~wolfram~~.com/~~BesselsInequality.html Bessel's Inequality]~~ the ~~article on Bessel's Inequality on MathWorld.~~

	~~{{PlanetMath attribution\|title=Bessel inequality\|id=3089}}~~

	~~{{Functional Analysis}}~~

	~~[[Category:Hilbert space]]~~
	~~[[Category:Inequalities]~~]

165.124.128.146: /* Software */

2014-01-23T21:24:25Z

Software

← Older revision		Revision as of 23:24, 23 January 2014
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	~~Greetings. Allow me begin by telling you the author~~'s ~~name - Phebe~~. ~~To gather coins~~ is one of the ~~things I adore most~~. ~~I am~~ a ~~meter reader~~. ~~Years~~ in the ~~past he moved to North Dakota~~ and ~~his family members loves it.~~<br><br>~~Feel free to visit my web~~-~~site~~ - ~~home std test kit~~ - [http://~~demo~~.~~crc-it~~.nl/~~content/candida-albicans-tips-you-need-remember read~~ article] -		In [[mathematics]], especially [[functional analysis]], '''Bessel's inequality''' is a statement about the coefficients of an element <math>x</math> in a [[Hilbert space]] with respect to an [[orthonormal]] [[sequence]].

			Let <math>H</math> be a Hilbert space, and suppose that <math>e_1, e_2, ...</math> is an orthonormal sequence in <math>H</math>. Then, for any <math>x</math> in <math>H</math> one has
			:<math>\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2 </math>

			where 〈•,•〉 denotes the [[inner product space\|inner product]] in the Hilbert space <math>H</math>. If we define the infinite sum
			:<math>x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, </math>
			consisting of 'infinite sum' of [[vector resolute]] <math>x</math> in direction <math>e_k</math>, Bessel's [[inequality (mathematics)\|inequality]] tells us that this [[series (mathematics)\|series]] [[Limit of a sequence\|converges]]. One can think of it that there exists <math>x' \in H</math> which can be described in terms of potential basis <math>e_1, e_2, ...</math>.

			For a complete orthonormal sequence (that is, for an orthonormal sequence which is a [[Orthonormal basis\|basis]]), we have [[Parseval's identity]], which replaces the inequality with an equality (and consequently <math> x'</math> with <math> x</math>).

			Bessel's inequality follows from the identity:
			:<math>0 \le \left\\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\\|^2 = \\|x\\|^2 - 2 \sum_{k=1}^n \|\langle x, e_k \rangle \|^2 + \sum_{k=1}^n \| \langle x, e_k \rangle \|^2 = \\|x\\|^2 - \sum_{k=1}^n \| \langle x, e_k \rangle \|^2,</math>
			which holds for any natural ''n''.

			==See also==
			* [[Cauchy–Schwarz inequality]]

			==External links==
			* {{springer\|title=Bessel inequality\|id=p/b015850}}
			* [http://mathworld.wolfram.com/BesselsInequality.html Bessel's Inequality] the article on Bessel's Inequality on MathWorld.

			{{PlanetMath attribution\|title=Bessel inequality\|id=3089}}

			{{Functional Analysis}}

			[[Category:Hilbert space]]
			[[Category:Inequalities]]

en>Sklynn at 20:52, 14 August 2012

2012-08-14T20:52:41Z

New page

Greetings. Allow me begin by telling you the author's name - Phebe. To gather coins is one of the things I adore most. I am a meter reader. Years in the past he moved to North Dakota and his family members loves it.<br><br>Feel free to visit my web-site - home std test kit - [http://demo.crc-it.nl/content/candida-albicans-tips-you-need-remember read article] -