H square

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

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n{e}^{in\phi }$

whereas elements of H² are given by

$\sum _{n=0}^{\mathrm{\infty }}{a}_n{e}^{in\phi }.$

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

On the half-plane

The Laplace transform ${ℒ}$ given by

$[{ℒ}f](s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}f(t)dt$

can be understood as a linear operator

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

where $L^2(0,\mathrm{\infty })$ 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

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

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

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

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.

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.