H Square

In mathematics and control theory, ''H''², or ''H-square'' is a Hardy space with square norm. It is a subspace of ''L''² space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.

On the unit circle

In general, elements of ''L''² on the unit circle are given by

:$\sum_^\infty a_n e^$

whereas elements of ''H''² are given by

:$\sum_^\infty a_n e^.$

The projection from ''L''² to ''H''² (by setting ''a''_''n'' = 0 when ''n'' < 0) is orthogonal.

On the half-plane

The Laplace transform $\mathcal$ given by

: mathcalfs)=\int_0^\infty e^f(t)dt

can be understood as a linear operator

:$\mathcal:L^2(0,\infty)\to H^2\left(\mathbb^+\right)$

where $L^2(0,\infty)$ is the set of square-integrable functions on the positive real number line, and $\mathbb^+$ 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

:$\, \mathcalf\, _ = \sqrt \, f\, _.$

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

:$L^2(\mathbb)=L^2(-\infty,0) \oplus L^2(0,\infty)$

one then obtains an orthogonal decomposition of $L^2(\mathbb)$ into two Hardy spaces

:$L^2(\mathbb)= H^2\left(\mathbb^-\right) \oplus H^2\left(\mathbb^+\right).$

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.

Control theory
Mathematical analysis