Convolution

A way to combine two functions, $(f*g)(t)=\int f(\tau)g(t-\tau)\,d\tau$ , that is neither their sum nor their product.

For functions $f,g$ on $\mathbb R$ , the convolution of $f$ and $g$ is $(f*g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau$ , defined wherever the integral converges. It is a natural way to combine two functions into a third, distinct from their sum, product, or composition -- turning multiplication in frequency into convolution in time and back, by the Convolution Theorem.

Examples

Convolving a signal with a narrow bump function smooths (blurs) it slightly, averaging each point with its close neighbors.