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Inmathematics, abasis function is an element of a particularbasis for afunction space. Every function in the function space can be represented as alinear combination of basis functions. In finite-dimensionalvector spaces this representation is purely algebraic and involves only finitely many basis functions, whereas in infinite-dimensional settings it typically takes the form of aninfinite series whose convergence depends on thetopology of the space.
Innumerical analysis andapproximation 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).
Themonomial basis for the vector space ofanalytic functions is given by
This basis is used inTaylor series, amongst others.
The monomial basis also forms a basis for the vector space ofpolynomials. After all, every polynomial can be written as for some, which is a linear combination of monomials.
Sines and cosines form an (orthonormal)Schauder basis forsquare-integrable functions on a bounded domain. As a particular example, the collectionforms a basis forL2[0,1].
