H Square - On The Half-plane

On The Half-plane

The Laplace transform given by

can be understood as a linear operator

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

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

$L^2(\mathbb{R})= H^2\left(\mathbb{C}^-\right) \oplus H^2\left(\mathbb{C}^+\right).$

This is essentially the Paley-Wiener theorem.