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 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.