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

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

whereas elements of H² are given by

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

The projection from L² to H² (by setting a_n = 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.

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.