Fourier’s transform trades time localization for frequency spread
The Fourier transform takes a function and returns another that measures how strongly each frequency is present, much as a chord can be split into the loudness of its separate pitches. Squeeze a signal in time and its spectrum spreads out, and a Gaussian transforms into another Gaussian.
Joseph Fourier introduced sine and cosine transforms while studying how heat flows, and his Analytical Theory of Heat also carried the inversion idea: given enough smoothness and decay, the original function can be rebuilt from its frequency portrait. Those sine and cosine pieces survive as the imaginary and real parts of the modern complex-valued version. Bell curves turned up naturally in his work because they solve the heat equation.
The trade-off between time and frequency is known as the uncertainty principle, and the Gaussian, familiar from probability, statistics, and diffusion, is its critical case. A single spike, the Dirac delta, transforms into a flat constant, spreading across every frequency at once. Handling such objects rigorously takes more than an improper Riemann integral; the delta and other tempered distributions are treated by duality instead.
Conventions vary. Measuring frequency in hertz gives a unitary form, while switching to angular frequency breaks the symmetry between the transform and its inverse unless a factor involving the square root of 2π is shared evenly between them. Whichever version is chosen, the signs in the forward and reverse exponentials must be opposite.
For absolutely integrable functions the result is bounded and uniformly continuous, and the Riemann–Lebesgue lemma says it fades to zero at infinity. On square-integrable functions the operation extends uniquely to a unitary operator. In several dimensions it carries position space to momentum space, which is why quantum mechanics relies on it. Replacing the real line with other groups yields the discrete-time transform, the discrete Fourier transform, and Fourier series for periodic functions, and the fast Fourier transform is the algorithm that computes the discrete version.
Source: Fourier transform
