A central concern of mathematical physics and engineering mathematics involves the transformation of equations into a coordinate system where expressions simplify, decouple, and are amenable to computation and analysis. Perhaps the most foundational and ubiquitous coordinate transformation was introduced by J.-B. Joseph Fourier in the early 1800s to investigate the theory of heat. Fourier introduced the concept that sine and cosine functions of increasing frequency provide an orthogonal basis for the space of solution functions. Indeed, the Fourier transform basis of sines and cosines serve as eigenfunctions of the heat equation, with the specific frequencies serving as the eigenvalues, determined by the geometry, and amplitudes determined by the boundary conditions.

Fourier's seminal work provided the mathematical foundation for Hilbert spaces, operator theory, approximation theory, and the subsequent revolution in analytical and computational mathematics. Fast forward two hundred years, and the fast Fourier transform has become the cornerstone of computational mathematics, enabling real-time image and audio compression, global communication networks, modern devices and hardware, numerical physics and engineering at scale, and advanced data analysis. Simply put, the fast Fourier transform has had a more significant and profound role in shaping the modern world than any other algorithm to date.

Fourier's seminal work provided the mathematical foundation for Hilbert spaces, operator theory, approximation theory, and the subsequent revolution in analytical and computational mathematics. Fast forward two hundred years, and the fast Fourier transform has become the cornerstone of computational mathematics, enabling real-time image and audio compression, global communication networks, modern devices and hardware, numerical physics and engineering at scale, and advanced data analysis. Simply put, the fast Fourier transform has had a more significant and profound role in shaping the modern world than any other algorithm to date.

### Section 2.1: Fourier Series and Transform

[ Fourier series ] [ Hilbert space ] [ Fourier transforms ] [Properties]

### Section 2.2: Discrete Fourier Transform