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hilbert_transform [2014/02/21 18:34] nikolaj |
hilbert_transform [2014/02/21 18:36] nikolaj |
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==== 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 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 $L^q(\mathbb R)$. |