The convolution theorem states that the Fourier transform or Laplace transform of the convolution integral of two functions f(t) and g(t) is equal to the product of the transforms of the functions. In other words,

f * g = \int_{-\infty}^{\infty} f(t - \tau) g(\tau) d \tau = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d \tau
\mathcal{F} \{ f * g \} = \mathcal{F} \{ f(t) \} \cdot \mathcal{F} \{ g(t) \}
\mathcal{L} \{ f * g \} = F(s)G(s).

In the case of a Laplace transform, since both g(t) and f(t) are assumed to be zero for t < 0, the convolution of f and g can also be written as

f * g = \int_0^t f(t - \tau) g(\tau) d \tau = \int_0^t f(\tau) g(t - \tau) d \tau.

It is also true that

\mathcal{F} \{ f  \cdot g \} = \mathcal{F} \{ f(t) \} * \mathcal{F} \{ g(t) \}
\mathcal{L} \{ f  \cdot  g \} = F(s) * G(s).

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