Mathematics

Fourier sketch

Sines that rebuild a periodic function, coefficients, spectra, and a discrete-transform note.

Basics

Decomposition

A decent periodic function can be rewritten as a constant plus sines and cosines at integer frequencies. A waveform with corners still matches if you add enough terms (Gibbs ringing at the jump). Harmonics in sound and circuits are the same picture. A finite record is treated as one period. At a jump the sum goes to the average of the two sides.

Coefficients

a_n and b_n are inner products (integrals) of the function with cosines and sines. Even functions often keep only cosines; odd only sines. The complex form c_n e^{i n ω t} is easier to compute. Parseval says time energy equals spectral energy. Truncating coefficients is a low-pass.

Spectrum

A line spectrum is the size of each harmonic. Aperiodic signals get a continuous spectrum via the Fourier transform. Windowing grows sidelobes. Sampling repeats the spectrum; past Nyquist it folds. An FFT is a fast algorithm for that discrete sum.

Waves and heat

The wave equation on a string and the heat equation on a rod have sine eigenmodes. Expand the initial shape in Fourier series; each mode oscillates at its frequency or decays exponentially. Fixed ends often want sines; insulated ends cosines. This page does not prove existence. It is a door into a PDE class.