Basis function

Inmathematics,abasis functionis an element of a particularbasisfor afunction space.Everyfunctionin the function space can be represented as alinear combinationof basis functions, just as every vector in avector spacecan be represented as a linear combination ofbasis vectors.

Innumerical analysisandapproximation theory,basis functions are also calledblending functions,because of their use ininterpolation:In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).

Examples

Monomial basis forC^ω

Themonomialbasis for the vector space ofanalytic functionsis given by $$\{x^{n}\mid n\in \mathbb {N} \}.$$

This basis is used inTaylor series,amongst others.

Monomial basis for polynomials

The monomial basis also forms a basis for the vector space ofpolynomials.After all, every polynomial can be written as$a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}$for some$n\in \mathbb {N}$,which is a linear combination of monomials.

Fourier basis forL²[0,1]

Sines and cosinesform an (orthonormal)Schauder basisforsquare-integrable functionson a bounded domain. As a particular example, the collection $$\{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}$$ forms a basis forL²[0,1].

See also

• Basis (linear algebra)(Hamel basis)
• Schauder basis(in aBanach space)
• Dual basis
• Biorthogonal system(Markushevich basis)
• Orthonormal basisin aninner-product space
• Orthogonal polynomials
• Fourier analysisandFourier series
• Harmonic analysis
• Orthogonal wavelet
• Biorthogonal wavelet
• Radial basis function
• Finite-elements (bases)
• Functional analysis
• Approximation theory
• Numerical analysis

References

• Itô, Kiyosi (1993).Encyclopedic Dictionary of Mathematics(2nd ed.). MIT Press. p. 1141.ISBN0-262-59020-4.