To obtain the Fourier transform of the BIE derived above, all quantities have to be extended from Ω to ℝ^{n} (the Fourier transformation is defined on ℝ^{n} and not on Ω). Formally, this can be done by defining a cutoff distribution χ which is simply one in the interior of Ω and zero outside. Then all quantities are multiplied by χ and finally transformed into Fourier space (windowed Fourier transform). Mathematically this extension and transformation is justified only in the frame of the theory of distributions.