Ref: [John B. Kogut, Rev. Mod. Phys. 51, 659 (1979), An introduction to lattice gauge theory and spin systems]. More precisely, please refer to Page 703 within the section of renormalization group analysis of Gaussian planar model in two dimensions.
When integrating the high momentum slicing for the Green function \begin{equation} G_{h}(x)=\int_{\Lambda-d\Lambda<p<\Lambda}\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\text{i}\mathbf{px}}}{p^{2}}=\int_{\Lambda-d\Lambda<p<\Lambda}\frac{pdpd\theta}{(2\pi)^{2}}\frac{e^{\text{i}{px}\cos\theta}}{p^{2}}=\frac{1}{(2\pi)^{2}}\int\frac{dp}{p}\int{e}^{\text{i}px\cos\theta}d\theta=\frac{1}{2\pi}\frac{d\Lambda}{\Lambda}J_{0}(\Lambda{x}), \end{equation} where in the last step the Bessel function is used to expand it. Note that the Bessel function does not fall rapidly as its argument increases. Then the author argued that through the so called smooth momentum space slicing procedure, $G_{h}(x)$ becomes a really short-ranged function. What does it mean and how to do it explicitly?