Consider the following convention for defining the Fourier transform
$\hat{f}(\omega) = \int f(x) e^{-2 \pi i x \omega } d\omega $.
Why is the Fourier transform of 1 equal to $\delta(\omega)$. Loosely, I see that when $\omega \neq 0$ the integral will be zero and when $\omega = 0$ the integral diverges but does anyone have a more rigorous way of showing this? Also, how do we know there is not some scaling factor in front or something?
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