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

edit

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 an = 0 when n < 0) is orthogonal.

On the half-plane

edit

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

edit

References

edit
  • 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.