The convolution integral formula is (f * g) ∧ (x) = (x) (x), and convolution is an important operation in analytical mathematics. Let f(x) and g(x) be two integrable functions on R 1 By integrating, it can be proved that the above integral exists for almost all x ∈ (-∞, ∞).

In this way, with the different values of x, this integral defines a new function h(x), which is called the convolution of F and G, and is recorded as h(x)=(f *g)(x). It is easy to verify that (f *g)(x)=(g *f)(x) and (f *g)(x) are still integrable functions.

Introduction:

Convolution is closely related to Fourier transform. If (x) and (x) respectively represent the Fourier transforms of F and G in L 1(R) 1, then the following relationship holds: (f * g) ∧ (x) = (x) (x), that is, the product of the Fourier transforms of the two functions is equal to the convolution Fourier transform. This relationship simplifies the handling of many problems in Fourier analysis.

The function (f *g)(x) obtained by convolution is generally smoother than F and G, especially when G is a smooth function with compact support and F is locally integrable, their convolution (f *g)(x) is also a smooth function. Using this property, for any integrable function? , we can simply construct a smooth function sequence fs(x) that approximates F. This method is called smoothing or regularization of functions.