FourierConvolutionTheorem - crowlogic/arb4j GitHub Wiki

Let's consider a function $h(x)$ defined as:

h(x) = \int_0^\infty f(x-y)g(y) \, dy

This is essentially a convolution of $f(x)$ and $g(y)$, often denoted as $f * g$. The convolution theorem states:

\mathcal{F}\{f * g\} = \mathcal{F}\{f\} \times \mathcal{F}\{g\}

where the Fourier transforms of $f(x)$ and $g(x)$ are given by:

F(k) = \int_{0}^{\infty} f(x) e^{-2\pi ixk} \, dx

G(k) = \int_{0}^{\infty} g(x) e^{-2\pi ixk} \, dx

Therefore, the Fourier transform of $h(x)$ is:

\mathcal{F}\{h(x)\} = F(k) \times G(k)

The original function $h(x)$ can be recovered through the inverse Fourier transform of this product:

h(x) = \mathcal{F}^{-1}\{ F(k) \times G(k) \}