I am trying to calculate the Fourier transform of of the function (distribution): \begin{cases} \delta(x-1) & x>0, \\ -\delta(-x-1) & x<0. \end{cases} I tried to rewrite it as $$\begin{align} & \operatorname{sign}(x)\cdot\delta(\operatorname{sign}(x)-1) = \\ &(2H(x)-1)\cdot\delta(2H(x)-2) \end{align} $$ but I do not se where to take it from here. Does it even make sense to use the Fourier transform on such a function? The Laplace transform would also be of some intrest to me. Help is appreciated!
2026-04-06 05:17:28.1775452648
Fourier Transform of $(2H(x)-1)\cdot\delta(2H(x)-2)$
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