Fourierkoefficienter

Fourierkoefficienter, or Fourier coefficients, are the numerical values that quantify the presence of each harmonic in a periodic function. Together they form the Fourier series, representing a periodic function as a sum of sinusoids or complex exponentials. For a function f with period T, the Fourier series can be written in the sine–cosine form

f(x) = a0/2 + sum_{n=1}^∞ [ a_n cos(2π n x / T) + b_n sin(2π n x / T) ],

where the coefficients are

a0 = (2/T) ∫_0^T f(x) dx,

a_n = (2/T) ∫_0^T f(x) cos(2π n x / T) dx,

b_n = (2/T) ∫_0^T f(x) sin(2π n x / T) dx.

An alternative complex form is

f(x) = sum_{n=-∞}^{∞} c_n e^{i 2π n x / T},

with

c_n = (1/T) ∫_0^T f(x) e^{−i 2π n x / T} dx.

Convergence and interpretation: if f is piecewise smooth (Dirichlet conditions), the series converges to the average of the left- and right-hand limits at discontinuities and to f(x) at points of continuity. Convergence can be nonuniform, and the Gibbs phenomenon may appear near jump discontinuities.

Computational aspects: for continuous data or sampled signals, coefficients can be estimated by numerical integration or by the discrete Fourier transform (DFT); in practice, the fast Fourier transform (FFT) is used to obtain many coefficients efficiently. Parseval’s identity relates the energy of the function over a period to the sum of the squared magnitudes of its coefficients.

Applications: Fourier coefficients are central to frequency analysis in signal processing, acoustics, and image analysis; they enable decomposition of signals, solution of partial differential equations with periodic boundary conditions, and characterization of a function’s harmonic content.