Mathematics 174 -- Applied Mathematics 2
Another Convolution Example
Consider the family of Gaussian functions
,
where
> g := (alpha,x) -> exp(-(x/alpha)^2)/(alpha*sqrt(Pi));
> plot({g(2,x),g(1,x),g(1/2,x),g(1/4,x)},x=-4..4,color=red);
As
-> 0+,
acquires a taller and taller peak at
x = 0,
while
for all
x
not
equal to zero. If we convolve with another function like
> f := x -> piecewise(abs(x) < 1, 1, 0);
> plot(f(x),x=-4..4,discont=true,color=black,thickness=3);
> convo := (alpha,x) -> int(f(t)*g(alpha,x-t),t=-1..1);
N.B. Limits of integration are
t = -
1..1
rather than
t =
, since
f(t)
is
zero outside the interval t = - 1..1 .
> plot(convo(1,x),x=-4..4);
> plot(convo(.25,x),x=-4..4);
> plot(convo(.05,x),x=-4..4);
>
Note that as
+, the graph of the convolution
f *
tends to approximate
f more and
more closely!
(This is not really a surprise: What is
as
?) However,
for all
, the convolution
f *
is infinitely differentiable (hence continuous).