Abstract: 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.

Key words:  convolution, smoothing, central limit theorem, differential equation, filtering