In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: 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
Polynomial bases
The collection of quadratic polynomials with real coefficients has {1, t, t^{2}} as a basis. Every quadratic polynomial can be written as a1+bt+ct^{2}, that is, as a linear combination of the basis functions 1, t, and t^{2}. The set {(t−1)(t−2)/2, −t(t−2), t(t−1)/2} is another basis for quadratic polynomials, called the Lagrange basis. The first three Chebyshev polynomials form yet another basis
Fourier basis
Sines and cosines form an (orthonormal) Schauder basis for squareintegrable functions. As a particular example, the collection:

\{\sqrt{2}\sin(2\pi n x) \;  \; n\in\mathbb{N} \} \cup \{\sqrt{2} \cos(2\pi n x) \;  \; n\in\mathbb{N} \} \cup\{1\}
forms a basis for L^{2}(0,1).
References

Ito, Kiyoshi (1993). Encyclopedic Dictionary of Mathematics (2nd ed. ed.). MIT Press. p. 1141.
See also
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