I'm trying to find the Fourier transform of $H(k - |x|)$, where $H$ is the Heaviside step function. I've solved a few Fourier transforms recently, but this one is giving me a bit of trouble. I'd appreciate any help.

  • 2
    $\begingroup$ $H(k-\lvert x\rvert)$ is the characteristic function of the ball with radius $k$, isn't it? Are we talking about $\mathbb{R}$ or $\mathbb{R}^n$? $\endgroup$ Dec 7 '14 at 15:08
  • $\begingroup$ I think just $\mathbb{R}$. I know that a sine should pop up in the answer. $\endgroup$
    – pidgeot
    Dec 7 '14 at 15:10
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    $\begingroup$ In that case $$\int_{-k}^k e^{-2\pi i \omega x}\,dx$$ or $$\frac{1}{\sqrt{2\pi}}\int_{-k}^k e^{-i\omega x}\,dx$$ can easily be evaluated directly, I'd say. It gets more complicated (Bessel functions) in higher dimensions. $\endgroup$ Dec 7 '14 at 15:13

As you said, there is a sinus behind:

$$\int_{-k}^ke^{−i2\pi\omega x}dx=\frac{e^{−i2\pi\omega k}-e^{i2\pi\omega k}}{-i2\pi\omega}=\frac{\sin 2\pi \omega k}{2\pi\omega}$$


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