Convolution of Functions: Definition, Geometric Interpretation, and Applications

This is the definition of the convolution of two functions 'a(t)' and 'b(t)'. It is denoted by 'a * b' and is defined as the sum of the product of 'a(τ)' and 'b(t-τ)' over all possible values of 'τ'.

Geometrically, convolution can be thought of as a way to combine two functions by 'sliding' one of them over the other and computing the area of overlap at each point. The resulting function represents the amount of overlap between the two functions at each point in time.

Convolution is a useful tool in many areas of mathematics and engineering, including signal processing, image processing, and probability theory. It has many important properties, such as associativity, distributivity, and commutativity, which make it a powerful tool for analyzing and manipulating functions.