H square

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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

  • 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.
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