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

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{in\varphi }}$

whereas elements of H² are given by

${\displaystyle \sum _{n=0}^{\infty }a_{n}e^{in\varphi }.}$

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

On the half-plane

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

${\displaystyle [{\mathcal {L}}f](s)=\int _{0}^{\infty }e^{-st}f(t)dt}$

can be understood as a linear operator

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

where ${\displaystyle L^{2}(0,\infty )}$ is the set of square-integrable functions on the positive real number line, and ${\displaystyle \mathbb {C} ^{+}}$ 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

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

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

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

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

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

This is essentially the Paley-Wiener theorem.

See also

• 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.