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Suppose f and g are two integrable functions on the interval (−π,π], and extended to be 2π-periodic on the real ner Then the convolution of f and g is the function defined for all x by: f∗g(x)=2π1∫−ππf(x−y)g(y)dy Example Let f be any function, and let g be defined as follows. Select a number 0<h<π. Then let g(x)={2π/h0 if −h/2≤x≤h/2 otherwise The function g is a tall narrow rectangle (if h is relatively small):
The function g is a tall narrow rectangle (if h is relatively small): Now the convolution of f and g is f∗g(x)=2π1∫−ππf(x−y)g(y)dy=2π1∫−h/2h/2f(x−y)(2π/h)dy=h1∫−h/2h/2f(x−y)dy
Now change variables: Let u=x−y. So du=−dy or −du=dy. If y=−h/2 then u=x−y=x−(−h/2)=x+h/2 If y=h/2 then u=x−y=x−h/2. This gives f∗g(x)=2π1∫−h/2h/2f(x−y)(1/h)dy=−h1∫x+h/2x−h/2f(u)du=h1∫x−h/2x+h/2f(u)du This is the average of the function f over the interval from x−h/2 to x+h/2. If x is the current time, this gives the average of over the interval of time of duration h, centered at x.