H square explained

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 ^{inf ty} _{n =-inf ty} a _{n} e ^{in\varphi}

whereas elements of H² are given by

\sum ^{inf ty} _{n =0} a _{n} e ^{in\varphi} .

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

On the half-plane

l{L} given by[ l{L}f](s)=\int ^{inf ty} _{0} e ^{-st} f(t)dt

can be understood as a linear operator

l{L}:L ^{2(0,infty)\to} H ^{2\left(C} ^{+\right)}

where

L ^{2(0,infty)} is the set of square-integrable functions on the positive real number line, and 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\| l{L}f\| _{H ^{2}} = \sqrt{ 2 \pi} \|f\| _{L ^{2}} .

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

L ^{2(R)=L} ^{2(-infty,0)} ⊕ L ^{2(0,infty)}

one then obtains an orthogonal decomposition of

L ^{2(R)} into two Hardy spacesL ^{2(R)= H} ^{2\left(C} ^{-\right)} ⊕ H ^{2\left(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, .