en>Hhhippo: ce, tag for context

2014-02-17T20:09:46Z

ce, tag for context

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	~~In [[mathematics]] and [[control theory]],~~ '~~''''H'''''<sup>2</sup>~~, or ''H-~~square'' is~~ a ~~[[Hardy space]] with square norm~~. It is ~~a subspace~~ of ~~[[Lp space\|''L''~~<~~sup~~>2<~~/sup~~> ~~space]], and is thus a [[Hilbert space]]~~. ~~In particular,~~ it is a ~~[[reproducing kernel Hilbert~~ space]].		They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. You may discover this probably the most time-consuming part of building a Word - Press MLM website. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. After activating, you will find their website link and get the activation code from their website. <br><br>Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. Wordpress have every reason with it which promote wordpress development. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Word - Press makes it possible to successfully and manage your website. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. Setting Up Your Business Online Using Free Wordpress Websites. To turn the Word - Press Plugin on, click Activate on the far right side of the list. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>If you have any concerns concerning in which and how to use [http://shortener.us/wordpress_backup_plugin_2603300 wordpress dropbox backup], you can get hold of us at our web site. Numerous bloggers are utilizing Word - Press and with good reason. But the Joomla was created as the CMS over years of hard work. Websites that do rank highly, do so becaue they use keyword-heavy post titles. There are many advantages of hiring Wordpress developers for Wordpress project development:. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. I don't want that for my visitors and I'm quite sure they don't either. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.

	~~== On the unit circle ==~~
	~~In general~~, ~~elements~~ of ~~''L''<sup>2</sup> on the unit circle are given by~~

	~~:<math>\sum_{n=~~-~~\infty}^\infty a_n e^{in\varphi}</math>~~

	~~whereas elements of ''H''<sup>2</sup> are given by~~

	~~:<math>\sum_{n=0}^\infty a_n e^{in\varphi}~~.~~</math>~~

	~~The projection from ''L''<sup>2</sup>~~ to ~~''H''~~<~~sup~~>2<~~/sup~~> ~~(by setting ''~~a~~''<sub>''n''</sub> = 0 when ''n'' < 0)~~ is ~~orthogonal~~.

	~~== On the half-plane ==~~
	The ~~[[Laplace transform]] <math>\mathcal{L}</math> given by~~

	~~:<math>[\mathcal{L}f](s)=\int_0^\infty e^{-st}f(t)dt</math>~~

	~~can be understood as~~ a ~~linear operator~~

	~~:<math>\mathcal{L}:L^2(0~~,~~\infty)\~~to
	~~H^2\left(\mathbb{C}^+\right)</math>~~

	~~where <math>L^2(0,\infty)</math> is~~ the ~~set of [[square~~-~~integrable]] functions~~ on ~~the positive real number line~~, ~~and <math>\mathbb{C}^+</math> is~~ the right ~~half~~ of the ~~complex plane~~. ~~It is more; it is an [[isomorphism]]~~, in that ~~it is invertible~~, and it [~~[isometry\|isometric]~~], in that ~~it satisfies~~

	~~:<math>\\|\mathcal{L}f\\|_{H^2} = \sqrt{2\pi} \\|f\\|_{L^2}~~.~~</math>~~

	~~The Laplace transform is "half"~~ of ~~a Fourier transform; from the decomposition~~

	:~~<math>L^2(\mathbb{R})=L^2(~~-~~\infty~~,~~0) \oplus L^2(0~~,~~\infty)</math>~~

	~~one then obtains an [[orthogonal decomposition]]~~ of <~~math~~>~~L^2(\mathbb{R})~~<~~/math~~> ~~into two Hardy spaces~~

	~~:<math>L^2(\mathbb{R})=~~
	~~H^2\left(\mathbb{C}^-\right) \oplus~~
	~~H^2\left(\mathbb{C}^+\right).</math>~~

	~~This~~ is ~~essentially the [[Paley-Wiener theorem]]~~.

	~~== See also==~~
	* [[H infinity\|''H'~~'<sub>∞</sub>]]~~

	~~==References==~~
	* Jonathan R. ~~Partington~~, ~~"Linear Operators and Linear Systems~~, ~~An Analytical Approach to Control Theory"~~, ~~''London Mathematical Society Student Texts '''60'''''~~, ~~(2004) Cambridge University Press~~, ~~ISBN 0~~-~~521~~-~~54619~~-2.

	~~[[Category:Control theory]]~~
	~~[[Category:Mathematical analysis]]~~

en>KT-2500 at 20:01, 25 September 2010

2010-09-25T20:01:41Z

New page

In [[mathematics]] and [[control theory]], '''''H'''''2, or ''H-square'' is a [[Hardy space]] with square norm. It is a subspace of [[Lp space|''L''2 space]], and is thus a [[Hilbert space]]. In particular, it is a [[reproducing kernel Hilbert space]].

== On the unit circle ==
In general, elements of ''L''2 on the unit circle are given by

:<math>\sum_{n=-\infty}^\infty a_n e^{in\varphi}</math>

whereas elements of ''H''2 are given by

:<math>\sum_{n=0}^\infty a_n e^{in\varphi}.</math>

The projection from ''L''2 to ''H''2 (by setting ''a''''n'' = 0 when ''n'' < 0) is orthogonal.

== On the half-plane ==
The [[Laplace transform]] <math>\mathcal{L}</math> given by

:<math>[\mathcal{L}f](s)=\int_0^\infty e^{-st}f(t)dt</math>

can be understood as a linear operator

:<math>\mathcal{L}:L^2(0,\infty)\to
H^2\left(\mathbb{C}^+\right)</math>

where <math>L^2(0,\infty)</math> is the set of [[square-integrable]] functions on the positive real number line, and <math>\mathbb{C}^+</math> is the right half of the complex plane. It is more; it is an [[isomorphism]], in that it is invertible, and it [[isometry|isometric]], in that it satisfies

:<math>\|\mathcal{L}f\|_{H^2} = \sqrt{2\pi} \|f\|_{L^2}.</math>

The Laplace transform is "half" of a Fourier transform; from the decomposition

:<math>L^2(\mathbb{R})=L^2(-\infty,0) \oplus L^2(0,\infty)</math>

one then obtains an [[orthogonal decomposition]] of <math>L^2(\mathbb{R})</math> into two Hardy spaces

:<math>L^2(\mathbb{R})=
H^2\left(\mathbb{C}^-\right) \oplus
H^2\left(\mathbb{C}^+\right).</math>

This is essentially the [[Paley-Wiener theorem]].

== See also==
* [[H infinity|''H''∞]]

==References==
* Jonathan R. Partington, "Linear Operators and Linear Systems, An Analytical Approach to Control Theory", ''London Mathematical Society Student Texts '''60''''', (2004) Cambridge University Press, ISBN 0-521-54619-2.

[[Category:Control theory]]
[[Category:Mathematical analysis]]

Photodetection - Revision history

en>Hhhippo: ce, tag for context

en>KT-2500 at 20:01, 25 September 2010