H square

In mathematics and control theory, '''''H'''''², or ''H-square'' is a Hardy space with square norm. It is a subspace of ''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''&nbsp;=&nbsp;0 when ''n''&nbsp;<&nbsp;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 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== * Hardy space * ''H''_&infin; * Unilateral shift operator

==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