Bump Function - Properties and Uses

Properties and Uses

While bump functions are smooth, they cannot be analytic unless they vanish identically. This is a simple consequence of the identity theorem.

Bump functions are often used as mollifiers, as smooth cutoff functions, and to form smooth partitions of unity. They are the most common class of test functions used in analysis.

The space of bump functions is closed under many operations. For instance, the sum, product, or convolution of two bump functions is again a bump function, and any differential operator with smooth coefficients, when applied to a bump function, will produce another bump function.

The Fourier transform of a bump function is a Schwartz function, but cannot be compactly supported unless it is zero, since the only entire analytic bump function is the zero function (see Paley–Wiener theorem). Because the bump function is infinitely differentiable, its Fourier transform F(k) must decay faster than any finite power of 1/k for a large angular frequency |k|. The Fourier transform of the particular bump function from above can be analyzed by a saddle-point method, and decays asymptotically as for large |k|.