Basisfunktion

Basisfunktion, in English commonly called a basis function, is a function that belongs to a basis of a function space and is used to express other functions in that space through linear combinations. A basis is a set of functions that is linearly independent and spans the space, so any function in the space can be represented as a (finite or infinite) sum of basis functions with corresponding coefficients.

In a finite-dimensional function space, every function f can be written as f = sum_i a_i φ_i, where φ_i are the basis functions and a_i are the expansion coefficients. In inner product spaces, choosing an orthonormal basis simplifies coefficient calculation: a_i = ⟨f, φ_i⟩. This yields a stable and straightforward projection of f onto the basis.

Basis functions come in various types. Orthogonal and orthonormal bases provide convenient coefficient computation, while non-orthogonal bases may require solving a linear system. Some bases have local support, such as the hat functions of finite element methods, leading to sparse representations; others are global, such as the trigonometric functions in Fourier bases, which represent smooth, periodic content efficiently.

Common examples include the Fourier basis (sine and cosine functions) on intervals like [-π, π], polynomial bases (monomials, Legendre polynomials) on finite domains, wavelet bases for localized analysis, and radial basis functions used in interpolation and certain machine learning methods. Basis functions underpin many techniques for function approximation, signal processing, numerical solution of differential equations, and data fitting.