Prove the convolution formula (4.5). Provide STEP-BY-STEP calculations.
The transform of convolutions.
The convolution of two functions f1(r) and f2(r) is defined as:
(f1 * f2)(r) = ∫f1(u)f2(r - u)du
By substitution, we can easily show that:
(f1 * f2)(r) = (f2 * f1)(r).
Through a straightforward calculation, one can prove that:
F(f1 * f2)(s) = √(2π)F(f1)(s)F(f2)(s) (4.5)