本文简介连续型和离散型卷积算子的概念、代数和分析性质以及卷积的几何意义解释；简述了其与傅里叶变换的关系和计算.特别地, 文中综述了卷积与其傅里叶变换在函数的平均值和光滑化、概率密度函数、中心极限定理、常微分方程求解、偏微分方程求解、滤波和图像处理中的应用.

In this paper, we introduce the basic concepts involving the convolution operators of discrete and continuous settings, review a few of algebraic and analytical properties of convolution, and give the geometric interpretation of convolution. The close relationship between convolution and Fourier transform is briefly recalled. As the applications of convolution operators, we study the mean value and smoothing, probability density function, central limit theorem, ordinary differential equation solution, and three kinds of typical partial differential equation by way of a combination of Fourier transform. Finally, the actions of convolution in general filtering and image processing are illustrated.