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I am not sure if you mean $x$ one or two dimensional variable. I don't have the article at hand but I guess we probably mean $D=1$?? The argument goes as follows: you have to use Bochner's theorem, which says that $W(x-y):=W_2(x,y)$ is positive distribution if and only if the Fourier transformation yields is a positive measure $\tilde W(p) d^Dp$. In this ...