cirkelbasis

Cirkelbasis (Dutch: circle basis) is a term used in mathematics to denote a basis for the space of functions defined on the circle S^1 that respects circular symmetry. In practice, it often refers to the Fourier basis, consisting of the complex exponentials e^{inθ} (n ∈ Z), or, in real form, the set {1, cos(nθ), sin(nθ)} (n ≥ 1). This basis is orthogonal in the L^2 sense with respect to the standard inner product on the circle. Any square-integrable function f(θ) on [0, 2π) can be expressed as a Fourier series f(θ) ~ ∑_{n=-∞}^{∞} c_n e^{inθ}, with coefficients c_n = (1/2π) ∫_0^{2π} f(θ) e^{-inθ} dθ.

Finite-dimensional approximations use a truncated set of modes, forming a finite-dimensional subspace called a circular basis of order N.

Applications include processing periodic signals, circular statistics, orientation data in robotics, and texture or pattern representation in computer graphics.

Relation: It is a special case of a Fourier basis on compact groups; on the circle S^1, the dual group is Z; the basis functions are the characters.

Terminology: The term cirkelbasis is not universally standardized; many texts simply refer to the Fourier basis on the circle.

See also: Fourier series, basis, L^2(S^1), harmonic analysis.