For the wave equation $u_{tt} = c^2\nabla^2 u$ with initial data $u(x,0) = f(x)$, $u_t(x,0) = g(x)$:

Taking Fourier transform in space: $\frac{\partial^2\hat{u}}{\partial t^2} = -4\pi^2c^2|\xi|^2\hat{u}$

Solution: $\hat{u}(\xi,t) = \hat{f}(\xi)\cos(2\pi c|\xi|t) + \frac{\hat{g}(\xi)}{2\pi c|\xi|}\sin(2\pi c|\xi|t)$

Inverse transforming gives the solution. The oscillatory nature in frequency space reflects wave propagation in physical space.

The linear Schrödinger equation $i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi$ becomes in Fourier space: $i\hbar\frac{\partial\hat{\psi}}{\partial t} = \frac{2\pi^2\hbar^2}{m}|\xi|^2\hat{\psi}$

giving $\hat{\psi}(\xi,t) = \hat{\psi}_0(\xi)e^{-2\pi^2i\hbar|\xi|^2t/m}$.

The dispersion relation $\omega = \frac{\hbar|\xi|^2}{2m}$ shows different frequencies travel at different speeds, causing wave packet spreading.

Computing discrete Fourier transforms naively requires $O(N^2)$ operations. The FFT algorithm reduces this to $O(N\log N)$ by recursively decomposing the transform: $\hat{f}_k = \sum_{j=0}^{N-1} f_j e^{-2\pi ijk/N}$

This computational breakthrough enables real-time signal processing and efficient numerical PDE solvers.

For the Dirichlet problem on the half-space $\{x_n > 0\}$: $\nabla^2 u = 0, \quad u(x', 0) = g(x')$

Taking Fourier transform in $x' = (x_1, \ldots, x_{n-1})$: $\frac{\partial^2\hat{u}}{\partial x_n^2} = 4\pi^2|\xi'|^2\hat{u}$

Solution: $\hat{u}(\xi', x_n) = \hat{g}(\xi')e^{-2\pi|\xi'|x_n}$

Inverse transforming gives the Poisson integral formula for the half-space.