I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider:
$$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = 0}^{N-1}\hat{v}_{p}\hat{u}_{p}e^{ik_{p}x_{m}}$$ where in $k_{p} = \frac{2\pi p}{L}$ and $m = 0,\ldots,N-1$
Then we can deduce that $ \hat{s}_{p} = \hat{v}_{p}\hat{u}_{p}$.
I can intuitively see how that deduction is made however I am unable to explain it. Can someone help me out?
Thanks in advance!