digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.

## On the unit circle

In general, elements of L2 on the unit circle are given by

$\sum_{n=-\infty}^\infty a_n e^{in\varphi}$

whereas elements of H2 are given by

$\sum_{n=0}^\infty a_n e^{in\varphi}.$

The projection from L2 to H2 (by setting an = 0 when n < 0) is orthogonal.

## On the half-plane

The Laplace transform $\mathcal{L}$ given by

$[\mathcal{L}f](s)=\int_0^\infty e^{-st}f(t)dt$

can be understood as a linear operator

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

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

$\|\mathcal{L}f\|_{H^2} = \sqrt{2\pi} \|f\|_{L^2}.$

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

$L^2(\mathbb{R})=L^2(-\infty,0) \oplus L^2(0,\infty)$

one then obtains an orthogonal decomposition of $L^2(\mathbb{R})$ into two Hardy spaces

$L^2(\mathbb{R})= H^2\left(\mathbb{C}^-\right) \oplus H^2\left(\mathbb{C}^+\right).$

This is essentially the Paley-Wiener theorem.

## 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.

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/H_square — Please support Wikipedia.
A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
 1000000 videos foundNext >
 1000000 videos foundNext >
 5 news items
 Sound Off: Your Predictions For Brock Lesnar Vs. Triple H At Extreme Rules ... Wrestling Inc. Wed, 15 May 2013 17:07:58 -0700 This Sunday at Extreme Rules, Brock Lesnar and Triple H square off for the third time, this time in a steel cage match. Who will win the rubber match? Will Paul Heyman get involved? Will CM Punk make an appearance? Will this be the last match between ... Semiconductor Today Semiconductor Today Tue, 07 May 2013 12:05:26 -0700 Plus, there's the news from organisations such as: 5N Plus, Aixtron, Amonix, Anadigics, ARC Energy, Ascent Solar, Avago, AWR, Bridgelux, Cree, CVD Equipment, Epistar, EV Group, Ferrotec-Temescal, First Solar, H-Square, Hitachi Cable, Infineon, IQE, ... maville.com Billes, jeux en bois et stratégie à la fête du jeu samedi en Sarthe maville.com Thu, 23 May 2013 07:00:21 -0700 Samedi de 14 h à 18 h, square des Ursulines et rue Jankowski, jeux de société, de réflexion, de stratégie et d'ambiance, jeux d'adresse géant. Un espace ludique « le Jardin ». Un espace petit enfance jusqu'à 3 ans. Un espace « éditeur » de jeux avec ... Signes : la foire bio fête ses 25 ans, ce dimanche Var-Matin Thu, 16 May 2013 02:20:29 -0700 Ouverture de 9 h à 19 h square Marcel-Pagnol. Entrée gratuite. Contacts : 04.94.57.94.30/ 04.94.41.71.05. Tweet. Tous droits de reproduction et de représentation réservés. © 2012 Agence France-Presse. Toutes les informations (texte, photo, vidéo, ... dans la ville la Nouvelle République Fri, 26 Apr 2013 20:33:00 -0700 68e anniversaire de la libération des camps de concentration et d'extermination. Dimanche, rassemblement à 11 h, square de la Résistance, rue des Martyrs. > Association Baobab-Aide enfants Burkina Faso. Dimanche, marche de solidarité de 4 km, 8 km ou ...
 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter