Wiener-Hopf

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The Wiener-Hopf technique was developed to solve Fourier transform problems with mixed conditions on the same boundary, by exploiting the complex-analytical properties of the transformed function.

[edit] Example

Let us consider the linear partial differential equation

$\boldsymbol{L}_{xy}f(x,y)=0,$

where $\boldsymbol{L}_{xy}$ is a linear operator which contains derivatives with respect to $x$ and $y$ , subject to the mixed conditions on $y = 0$ , for some prescribed function $g (x)$ ,

$f = g (x)$ for $x\geq 0,$

$f y = 0$ when $x < 0$ .

and decay at infinity i.e. $f\rightarrow 0$ as $\boldsymbol{x}\rightarrow \infty$ . Taking a Fourier transform with respect to x results in the following ODE

$\boldsymbol{L}_{y}\hat{f}(k,y)-P(k,y)\hat{f}(k,y)=0,$

where $\boldsymbol{L}_{y}$ is a linear operator containing $y$ derivatives only, $P (k, y)$ is a known function of $y$ and $k$ and

$\hat{f}(k)=\int_{-\infty}^{\infty} f(x,y)e^{-ikx}\textrm{d}x.$

If a particular solution of the ODE which satisfies the necessary decay at infinity is denoted $\hat{F}(k,y)$ , a general solution can be written as

$\hat{f}=C(k)\hat{F}(k,y)$

where $C (k)$ is an unknown function to be determined by the boundary conditions on $y = 0$ .

The key idea is to split $\hat{f}$ into two separate functions, $\hat{f}_{+}$ and $\hat{f}_{-}$ which are analytic in the lower- and upper-halves of the complex plane, respectively

$\hat{f}_{+}(k,y)=\int_{0}^{\infty} f(x,y)e^{-ikx}\textrm{d}x,$
$\hat{f}_{-}(k,y)=\int_{-\infty}^{0} f(x,y)e^{-ikx}\textrm{d}x.$

The boundary conditions then give

$\hat{f}_{-}(k,0)+\hat{f}_{+}(k,0) = \hat{g}(k)+\hat{f}_{+}(k,0) = C(k)F(k,0)$

and, on taking derivatives with respect to $y$ ,

$\hat{f}'_{-}(k,0)+\hat{f}'_{+}(k,0) = \hat{f}'_{-}(k,0) = C(k)F'(k,0).$

Eliminating $C (k)$ yields

$\hat{g}(k)+\hat{f}_{+}(k,0) - \hat{f}'_{-}(k,0)/K(k)$ = 0,

where

$K(k) = \frac{F'(k,0)}{F(k,0)}.$

Now $K (k)$ can be decomposed into the product of functions $K -$ and $K +$ which analytic in the upper-half plane or lower-half plane, respectively

$K (k) = K + (k) K - (k),$

$\hbox{log} K^{-} = \frac{1}{2\pi i}\int_{-\infty}^{\infty} \frac{\hbox{log}(K(z))}{z-k} \textrm{d}z, \quad \hbox{Im}k>0,$
$\hbox{log} K^{+} = -\frac{1}{2\pi i}\int_{-\infty}^{\infty} \frac{\hbox{log}(K(z))}{z-k} \textrm{d}z, \quad \hbox{Im}k<0.$

Consequently

$K_{+}(k)\hat{g}_{+}(k) + K_{+}(k)\hat{f}_{+}(k,0) = \hat{f}'_{-}(k,0)/K_{-}(k) - K_{+}(k)\hat{g}_{-}(k),$

where it has been assumed that $g$ can be broken down into functions analytic in the lower-half plane $g +$ and upper-half plane $g -$ , respectively. Now, as the left-hand side of the above equation is analytic in the lower-half plane, whilst the right-hand side is analytic in the upper-half plane, analytic continution guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large $k$ , Liouville's theorem (complex analysis) tells us that this entire function is identically zero, therefore

$\hat{f}_{+}(k,0) = -\hat{g}_{+}(k),$