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

[math]\displaystyle{ \sum_{n=-\infty}^\infty a_n e^{in\varphi} }[/math]

whereas elements of H² are given by

[math]\displaystyle{ \sum_{n=0}^\infty a_n e^{in\varphi}. }[/math]

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

On the half-plane

The Laplace transform [math]\displaystyle{ \mathcal{L} }[/math] given by

[math]\displaystyle{ [\mathcal{L}f](s)=\int_0^\infty e^{-st}f(t)dt }[/math]

can be understood as a linear operator

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

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

[math]\displaystyle{ \|\mathcal{L}f\|_{H^2} = \sqrt{2\pi} \|f\|_{L^2}. }[/math]

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

[math]\displaystyle{ L^2(\mathbb{R})=L^2(-\infty,0) \oplus L^2(0,\infty) }[/math]

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

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

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.