Summary: Convolutions and Fourier Transforms.
f : R R.
We say f is summable if f is integrable over each bounded interval (whatever that means) and
|f(x)| dx < .
Definition. Suppose f and g are summable. Then
f g(x) =
f(x - y)g(y) dy
is defined and continous for all x (althought this is not immediately obvious) and f g is summable. In fact
||f g|| =
|f g(x)| dx