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

whereas elements of H² are given by

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

On the half-plane

The Laplace transform given by

can be understood as a linear operator

where is the set of square-integrable functions on the positive real number line, and 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

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

one then obtains an orthogonal decomposition of into two Hardy spaces

This is essentially the Paley-Wiener theorem.

See also

• Hardy space
• H_&infin;
• Unilateral shift operator

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, .