# Differences

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 hilbert_transform [2014/02/21 18:31]nikolaj hilbert_transform [2014/02/21 18:36]nikolaj 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 2: Line 2: ==== Partial Function ==== ==== Partial Function ==== | @#FFBB00: $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ | | @#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: $H(f):=y\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(x)}{y-x}\,\mathrm dx$ | ==== Discussion ==== ==== Discussion ==== - $H(H(f))=-f$ + $(H(f))=-f$ - The Hilbert transform is used in the Kramers–Kronig relation/​Sokhotski–Plemelj theorem to express the imaginary part of an analytic function in terms of its real part (or the other way around). This works because they get "mixed up" in the Cauchy integral formula which introduces a factor of $\frac{1}{i}$. The principal value is taken to push the complex line integral on the real line. + 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 $L^q(\mathbb R)$. + + It is also used in the Kramers–Kronig relation/​Sokhotski–Plemelj theorem to express the imaginary part of an analytic function in terms of its real part (or the other way around). This works because they get "mixed up" in the Cauchy integral formula which introduces a factor of $\frac{1}{i}$. The principal value is taken to push the complex line integral on the real line. === Reference === === Reference === Wikipedia: [[http://​en.wikipedia.org/​wiki/​Hilbert_transform|Hilbert transform]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Hilbert_transform|Hilbert transform]] ==== Parents ==== ==== Parents ==== + === Subset of === + [[Bounded linear operator]] === Requirements === === Requirements === [[Cauchy principal value]] [[Cauchy principal value]] 