# Differences

This shows you the differences between two versions of the page.

 hilbert_transform [2014/02/21 18:34]nikolaj hilbert_transform [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2014/02/21 18:36 nikolaj 2014/02/21 18:34 nikolaj 2014/02/21 18:31 nikolaj 2014/02/21 18:22 nikolaj 2014/02/21 18:22 nikolaj 2014/02/21 18:21 nikolaj 2014/02/21 18:21 nikolaj old revision restored (2014/02/21 18:16) Next revision Previous revision 2014/02/21 18:36 nikolaj 2014/02/21 18:34 nikolaj 2014/02/21 18:31 nikolaj 2014/02/21 18:22 nikolaj 2014/02/21 18:22 nikolaj 2014/02/21 18:21 nikolaj 2014/02/21 18:21 nikolaj old revision restored (2014/02/21 18:16) Line 1: Line 1: ===== Hilbert transform ===== ===== Hilbert transform ===== ==== Partial Function ==== ==== Partial Function ==== - | @#FFBB00: $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ | + | @#FFBB00: definiendum ​| @#FFBB00: $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ | - | @#FFBB00: $H(f):=t\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(\tau)}{t-\tau}\,\mathrm dx$ | + | @#FFBB00: definiendum ​| @#FFBB00: $H(f):=y\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(x)}{y-x}\,\mathrm dx$ | ==== Discussion ==== ==== Discussion ==== - $H(H(f))=-f$ + $(H(f))=-f$ The Hilbert transform commutes with the [[Fourier transform]] up to a simple factor and is an anti-self adjoint operator relative to the duality pairing between $L^p(\mathbb R)$ and the dual space $L^q(\mathbb R)$. The Hilbert transform commutes with the [[Fourier transform]] up to a simple factor and is an anti-self adjoint operator relative to the duality pairing between $L^p(\mathbb R)$ and the dual space $L^q(\mathbb R)$. 