H square

In general, elements of L2 on the unit circle are given by :\sum_{n=-\infty}^\infty a_n e^{in\varphi} whereas elements of H2 are given by :\sum_{n=0}^\infty a_n e^{in\varphi}. The projection from L2 to H2 (by setting an = 0 when n < 0) is orthogonal. == On the half-plane ==

The Laplace transform \mathcal{L} given by :[\mathcal{L}f](s)=\int_0^\infty e^{-st}f(t)dt can be understood as a linear operator :\mathcal{L}:L^2(0,\infty)\to H^2\left(\mathbb{C}^+\right) where L^2(0,\infty) is the set of square-integrable functions on the positive real number line, and \mathbb{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 :\|\mathcal{L}f\|_{H^2} = \sqrt{2\pi} \|f\|_{L^2}. The Laplace transform is "half" of a Fourier transform; from the decomposition :L^2(\mathbb{R})=L^2(-\infty,0) \oplus L^2(0,\infty) one then obtains an orthogonal decomposition of L^2(\mathbb{R}) 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. == See also==