In functional analysis, Convolution, convolution or convolution (English: convolution) is a mathematical operator that generates a third function from two functions F and G, representing the accumulation of the overlapping part of function F and G after inversion and translation. If the function involved in convolution is regarded as interval indicator function, convolution can also be regarded as a generalization of "moving average".

Briefly introduce the definition of convolution.

Convolution is an important operation in analytical mathematics. Let f (x) and g (x) be two integrable functions on R 1 and make an integral (as shown on the right): it can be proved that the above integral exists for almost all real numbers x, so that with the different values of x, this integral defines a new function h(x), which is called the convolution of functions f and g, and is recorded as h (x) =. It is easy to verify that (f * g)(x) = (g * f)(x) and (f * g)(x) are still integrable functions. That is to say, the space L 1(R 1) 1 is not a multiplication, but an algebra, even a Banach algebra. Convolution is closely related to Fourier transform. Many problems in Fourier analysis can be simplified by using the property that the product of Fourier transform of two functions is equal to the convolution Fourier transform. The function f*g obtained by convolution is generally smoother than f and g, especially when g is a smooth function with compact set and f is locally integrable, their convolution f * g is also a smooth function. Using this property, a series of smooth function sequences fs can be simply constructed for any integrable function f, which is called smoothing or regularization of functions. The concept of convolution can also be extended to sequences, measures and generalized functions.