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.

See also

References

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