Convolution Theorem

Convolution Theorem: Convolution Theorem are given as:

If

Proof :

To prove (1) it is sufficient to prove that;

The domain of integration for the above double integral is from u = 0 to u = t and t = 0 to t = infinite. The double integral given in the RHS of equation (3) indicates that we integrate first parallel to u-axis and then parallel to t-axis. We shall now change the order of integration parallel to t-axis the limits being t = u to t = infinite and parallel to u-axis the limit being u = 0 to infinite.

∴ From equation (3) we get;