Mathematics 300 -- Reading Course in Real Analysis

April 16, 1999

A Convolution Example --

We let f(x) = 1 if ,

0 if

(a "boxcar" function) and g(x) be the Gaussian: . The convolution f * g is

the function ( f * g)(x) = = , or as a Maple function:

> conv:=x->evalf(int((10/sqrt(Pi))*exp(-100*(x-y)^2),y=-1..1));

Here's the plot of the convolution function:

> plot(conv(x),x=-2..2);

Note that this essentially looks like a "smeared-out" version of the boxcar function f, with the

discontinuities removed. (In fact, this function is infinitely differentiable!)

For each fixed x, we can visualize the value of ( f * g)(x) like this: the graph of g(x - y) is a shifted

and reflected copy of the graph of g(y). Hence the integral of f(y)g(x - y) is (for this f) the portion

the area under the graph of g(x - y) that lies between y = -1 and y = 1. CAUTION: you want

to think of BOTH x and y as horizontal axes here.

> g:=x->10*exp(-100*x^2)/sqrt(Pi);

> with(plots):

> animate(g(x-y),y=-1..1,x=-1.2..1.2,numpoints=100,frames=50);

At the x-value represented by the peak (max) of the blue curve as it moves, the convolution

is equal to the area under the curve between -1 and 1.

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