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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 $L^q(\mathbb R)$.
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