Fourier transforms

From series to transform. Convolution, Parseval, uncertainty.

Fourier transforms decompose a function into its frequency content. The same operation diagonalizes every translation-invariant linear operator on $\mathbb{R}^n$ , which is why the toolkit shows up in nearly every branch of physics.Heat equation, wave equation, Schrödinger equation, scattering theory, signal processing, optics. The transform is the lingua franca of linear PDEs.

\tilde{f}(k) = \int_{-\infty}^{\infty} f(x)\,e^{-ikx}\,dx, \qquad f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde{f}(k)\,e^{ikx}\,dk.

Placeholder. Full chapter coming.