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en>Niceguyedc: WPCleaner v1.33 - Repaired 1 link to disambiguation page - (You can help) - Mirror (computing)

2014-08-19T01:12:55Z

WPCleaner v1.33 - Repaired 1 link to disambiguation page - (You can help) - Mirror (computing)

← Older revision		Revision as of 03:12, 19 August 2014
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	~~In [[mathematics]], a '''de Branges space''' (sometimes written '''De Branges space''')~~ is ~~a concept in [[functional analysis]] and is constructed from a '~~'~~'de Branges function'''.~~		Marvella is what you can contact her but it's not the most feminine name out there. For years I've been operating as a payroll clerk. What I love doing is taking part in baseball but I haven't made a dime with it. For a whilst she's been in South Dakota.<br><br>Check out my website - std home test ([http://fankut.com/index.php?do=/profile-2572/info/ click the up coming site])

	~~The concept is named after [[Louis de Branges]] who proved numerous results regarding these spaces, especially as [[Hilbert space]]~~s~~, and used those results to prove~~ the ~~[[Bieberbach conjecture]]~~.
	~~==De Branges functions==~~
	~~A '''de Branges function''~~' is ~~an [[entire function]]~~ ''E'' from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> that satisfies the inequality <math>\|E(z)\| > \|E(\bar z)\|</math>, for all ''z'' in the [[upper half-plane\|upper half of the complex plane]] <math>\mathbb{C}^+ = \{z \in \mathbb{C} \| {\rm Im}(z) > 0\}</math>.

	~~==Definition 1==~~
	~~Given~~ a ~~de Branges function ''E'', the de Branges space ''B''(''E'~~'~~) is defined as the set of all [[entire function]]~~s ~~''F'' such that~~

	~~: <math>F/E,F^{\#}/E \~~in ~~H_2(\mathbb{C}^+)~~<~~/math~~>

	~~where:~~

	* <~~math~~>~~\mathbb{C}^+ = \{z \in \mathbb{C} \| {\rm Im~~(~~z)} > 0\}<~~/~~math> is the open upper half of the complex plane.~~
	* <math>F^{\#}(z) = \overline{F(\bar z)}</~~math>~~.
	* <math>H_2(\mathbb{C}^+)</~~math> is the usual [[Hardy space]] on the open upper half plane~~.

	=~~=Definition 2==~~
	~~A de Branges space can also be defined as all entire functions ''F'' satisfying all of the following conditions:~~

	* <math>\int_{\mathbb{R}} \|(F/E)(\lambda)\|^2 d\lambda < \infty </math>
	* <math>\|(F/~~E)(z)\|,\|(F^{\#}/E)(z)\| \leq C_F(\operatorname{Im}(z))^{(~~-1/~~2)}, \forall z \in \mathbb{C}^+<~~/~~math>~~

	~~==As Hilbert spaces==~~
	~~Given a de Branges space ''B''(''E''). Define~~ the ~~scalar product:~~

	~~:<math>[F,G~~]~~=\frac{1}{\pi} \int_{\mathbb{R}} \overline{F(\lambda~~)~~} G(\lambda) \frac{d\lambda}{\|E(\lambda)\|^2}.</math>~~

	~~A de Branges space with such a scalar product can be proven to be a [[Hilbert space]].~~

	~~==References==~~

	* {{cite journal\|author=Christian Remling\|title=Inverse spectral theory for one-dimensional Schrödinger operators: the A function\|journal=Math. Z.\|volume=245\|year=2003}}

	~~[[Category:Operator theory]]~~
	~~[[Category:Hardy spaces]]~~

77.56.53.183: /* Theoretical models */

2014-02-03T21:03:48Z

Theoretical models

← Older revision		Revision as of 23:03, 3 February 2014
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	~~Irwin Butts~~ is ~~what my wife enjoys to call me although I don~~'~~t really like being known~~ as ~~like that~~. ~~One of~~ [~~http:~~//~~mcb-law.net~~/~~candidiasis~~-~~tips-you-have-to-remember/ over~~ the ~~counter std test~~] ~~very very best things~~ in the ~~world for me~~ is ~~to do~~ [~~http~~://~~www.animecontent.com~~/~~blog/436250 at home std testing] std test aerobics and now I'm [http~~:/~~/payiz~~.az/~~index~~.~~php?do=~~/~~profile-14379/info/ std home test~~] ~~trying to make cash with it~~. ~~Since she was 18 she~~'~~s been working as a meter reader but she~~'~~s usually needed her own [http~~://~~Www.rxlist.com~~/~~collection-of~~-~~images~~/~~genital_warts_hpv_picture~~/~~pictures~~.~~htm business~~]. [~~http~~:~~//www~~.~~Mayoclinic~~.~~org/diseases-conditions/sexually-transmitted-diseases-stds/in-depth/std-testing/art-20046019 South Dakota~~] ~~is exactly where I've usually been residing.~~		In [[mathematics]], a '''de Branges space''' (sometimes written '''De Branges space''') is a concept in [[functional analysis]] and is constructed from a '''de Branges function'''.

			The concept is named after [[Louis de Branges]] who proved numerous results regarding these spaces, especially as [[Hilbert space]]s, and used those results to prove the [[Bieberbach conjecture]].
			==De Branges functions==
			A '''de Branges function''' is an [[entire function]] ''E'' from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> that satisfies the inequality <math>\|E(z)\| > \|E(\bar z)\|</math>, for all ''z'' in the [[upper half-plane\|upper half of the complex plane]] <math>\mathbb{C}^+ = \{z \in \mathbb{C} \| {\rm Im}(z) > 0\}</math>.

			==Definition 1==
			Given a de Branges function ''E'', the de Branges space ''B''(''E'') is defined as the set of all [[entire function]]s ''F'' such that

			: <math>F/E,F^{\#}/E \in H_2(\mathbb{C}^+)</math>

			where:

			* <math>\mathbb{C}^+ = \{z \in \mathbb{C} \| {\rm Im(z)} > 0\}</math> is the open upper half of the complex plane.
			* <math>F^{\#}(z) = \overline{F(\bar z)}</math>.
			* <math>H_2(\mathbb{C}^+)</math> is the usual [[Hardy space]] on the open upper half plane.

			==Definition 2==
			A de Branges space can also be defined as all entire functions ''F'' satisfying all of the following conditions:

			* <math>\int_{\mathbb{R}} \|(F/E)(\lambda)\|^2 d\lambda < \infty </math>
			* <math>\|(F/E)(z)\|,\|(F^{\#}/E)(z)\| \leq C_F(\operatorname{Im}(z))^{(-1/2)}, \forall z \in \mathbb{C}^+</math>

			==As Hilbert spaces==
			Given a de Branges space ''B''(''E''). Define the scalar product:

			:<math>[F,G]=\frac{1}{\pi} \int_{\mathbb{R}} \overline{F(\lambda)} G(\lambda) \frac{d\lambda}{\|E(\lambda)\|^2}.</math>

			A de Branges space with such a scalar product can be proven to be a [[Hilbert space]].

			==References==

			* {{cite journal\|author=Christian Remling\|title=Inverse spectral theory for one-dimensional Schrödinger operators: the A function\|journal=Math. Z.\|volume=245\|year=2003}}

			[[Category:Operator theory]]
			[[Category:Hardy spaces]]

213.68.144.90: /* File-based solutions */

2012-08-24T06:54:58Z

File-based solutions

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