I have two sequences of the same length, $(x_i), i=1, 2, \ldots, N$ and $(y_i), i=1, 2, \ldots, N$ and a function $K(t) = -t \times \exp(-t^2 / 2)/ \sqrt{2 \pi}$.
I need to compute the following quantity for each $m=1, 2, \ldots, N$:
$\sum_{j=1}^N K(x_m - x_j) \times y_j$
which is a tad slow when done directly (I need it when $N = O(10^4)$ ). I know this can be much improved on with the use of FFT, however it is something I never really worked with. Could anyone suggest any links or a way to rewrite it in FFT form?
I'd be very grateful for any help :)