MATH201/content/Convolution (lec 19).md

Convolution

A convolution is an operation of function, we take two functions, convolute them and get a new function. Definition of convolution between f and g:

(f*g)(t):=\int _{0} ^t f(t-v)g(v)\, dv

property 1) f*g=g*f proof: f*g=\int _{0} ^t f(t-v)g(v)\, \underset{ t-v=u }{ dv }=-\int _{t} ^0 f(u)g(t-u) \, du =\int _{0} ^t g(t-u)f(u)\, du=g*f \quad \Box

property 2) (f+g)*h=f*h+g*h property 3) (f*g)*h=f*(g*h) property 4) f*0=0 property 5) \mathcal{L}\{f*g\}=F(s)G(s) he will see us tomorrow at 10oclock. ;) #end of lec 19