A Fourier series is a mathematical expansion of periodic functions into a sum of sine and cosine waves.A Fourier series is a representation of a periodic function f(x) with period T as a sum of trigonometric functions: f(x) = a0/2 + Σ[a_n cos(2π n x/T) + b_n sin(2π n x/T)]. The coefficients a_n and b_n are calculated as projections of the function onto the corresponding cosine and sine waves and show how strongly each harmonic is expressed in the original signal. The expansion is possible due to the orthogonality of sines and cosines. Fourier series are a powerful tool for solving differential equations (e.g., the heat equation) and underpin spectral analysis.
The origins of the idea go back to Daniel Bernoulli (1753), who represented string motion as a sum of harmonics. Jean-Baptiste Joseph Fourier in 1807, in his memoir on heat conduction, systematically used trigonometric series and proposed a method for computing coefficients. His work 'The Analytical Theory of Heat' (1822) established this apparatus, despite criticism from contemporaries for lack of rigor. A rigorous convergence theory was built by Peter Gustav Lejeune Dirichlet in 1829, and later developed by Riemann, Lebesgue, and others.
When expanding f(x), harmonics with numbers n form a spectrum. Low n correspond to slow changes, high n to fast details. The integrals for coefficients a_n and b_n are the scalar product of the function and the cosine/sine; thanks to the orthogonality of the basis ∫ cos(nx) cos(mx) = 0 for n≠m, each harmonic 'picks out' only its part of the signal. The sum of all harmonics gives the original function almost everywhere, and when truncated to a finite number, ensures minimum mean square error. The Fast Fourier Transform exploits the symmetry of complex roots of unity, recursively reducing the computation of an N-point DFT to two N/2-point ones, dramatically accelerating processing. In physics, harmonic decomposition is used for time series analysis (astronomy), mode composition of laser beams, plasma diagnostics, and solving Schrödinger equations.
Audio and image compression (MP3, JPEG), vibration analysis in structures, seismic data processing, and quantum computing.
When expanding a rectangular signal into a Fourier series, the sum of all odd harmonics yields a staircase that approaches a rectangle as the number of terms increases, but near the discontinuity, 'bumps' of about 9% of the signal amplitude remain — this is the Gibbs phenomenon, occurring in any series with discontinuities.
| \tilde{\kappa} | variable | Fourier transform of convergence — convergence |
| \tilde{\kappa}_B | variable | Fourier transform of B-mode convergence — b mode convergence |
| \tilde{\gamma}_1 | variable | Fourier transform of first shear component — shear component |
| \tilde{\gamma}_2 | variable | Fourier transform of second shear component — shear component |
| k_1 | variable | First component of wave vector — wave vector component radian per metre |
| k_2 | variable | Second component of wave vector — wave vector component radian per metre |
| k | variable | Magnitude of wave vector — wave number radian per metre |
| x_t | variable | value of the process at time t — time series value |
| x_{t-1} | variable | value of the process at previous time — time series value |
| \phi | variable | autoregressive coefficient — autoregressive coefficient |
| \epsilon_t | variable | white noise innovation at time t — noise term |
| \sigma^2 | variable | variance of the noise — variance |
