basisfunction

A basis function is a member of a set of functions used to construct other functions in a function space. If a set of basis functions {phi_i} spans a space V, every function f in V can be written uniquely as a linear combination f = sum_i c_i phi_i, with coefficients c_i determined by the representation. The idea generalizes the standard basis of Euclidean space, where basis vectors correspond to coordinate axes.

In finite-dimensional spaces, the basis contains a finite number of functions equal to the dimension of the space. In function spaces, common choices of basis functions include global or localized families such as Fourier modes (sine and cosine functions), Legendre or Chebyshev polynomials, wavelets, splines, and piecewise polynomial shape functions. These choices affect convergence, efficiency, and suitability for the problem at hand.

Basis functions are central to numerical approximation and analysis. In the finite element method, for example, piecewise polynomial basis (shape) functions are associated with nodes or elements and used to approximate solutions to differential equations. In spectral methods, orthogonal bases like trigonometric or polynomial families enable high-accuracy representations. Orthogonality often simplifies computation: coefficients can be obtained by inner products, and orthonormal bases yield direct projection formulas.

Variants and terminology include orthonormal bases in Hilbert spaces, and Schauder bases in more general Banach spaces, where convergence considerations replace simple finite sums. The concept of a basis function thus underpins many techniques for representing, approximating, and analyzing functions across mathematics, physics, and engineering.