domain: The Fourier transform of a bump function is a (real) analytic function, and it can be extended to the whole complex plane: hence it cannot be compactly supported unless it is zero, since the only entire analytic bump function is the zero function (see Paley–Wiener theorem and Liouville's theorem). Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of for a large angular frequency . The Fourier transform of the particular bump function from above can be analyzed by a saddle-point method, and decays asymptotically as for large . See also Citations References Smooth functions