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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, a `!basis function`! is an element of a particular `F33f`_`[basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Basis_(linear_algebra)]`_`f for a `F33f`_`[function space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_space]`_`f. Every `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f in the function space can be represented as a `F33f`_`[linear combination`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_combination]`_`f of basis functions, just as every vector in a `F33f`_`[vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f can be represented as a linear combination of `F33f`_`[basis vectors`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Basis_vectors]`_`f.

In `F33f`_`[numerical analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Numerical_analysis]`_`f and `F33f`_`[approximation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Approximation_theory]`_`f, basis functions are also called `!blending functions,`! because of their use in `F33f`_`[interpolation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Interpolation]`_`f: 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).

>>Contents

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Monomial basis for C ω`#monomial-basis-for-c]`_`f

• `F0af`_`[Monomial basis for polynomials`#monomial-basis-for-polynomials]`_`f

• `F0af`_`[Fourier basis for L 2 [0,1`#fourier-basis-for-l-2-0-1]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Examples

>>>Monomial basis for C ω

The `F33f`_`[monomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monomial]`_`f basis for the vector space of `F33f`_`[analytic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Analytic_function]`_`f is given by { x n ∣ ∣ n ∈ ∈ N } . {\\displaystyle \\{x^{n}\\mid n\\in \\mathbb {N} \\}.}

This basis is used in `F33f`_`[Taylor series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_series]`_`f, amongst others.

>>>Monomial basis for polynomials

The monomial basis also forms a basis for the vector space of `F33f`_`[polynomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial]`_`f. After all, every polynomial can be written as a 0 + a 1 x 1 + a 2 x 2 + ⋯ ⋯ + a n x n {\\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\\cdots +a_{n}x^{n}} for some n ∈ ∈ N {\\displaystyle n\\in \\mathbb {N} } , which is a linear combination of monomials.

>>>Fourier basis for L 2 [0,1]

`F33f`_`[Sines and cosines`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Trigonometric_functions]`_`f form an (`F33f`_`[orthonormal`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthonormality]`_`f) `F33f`_`[Schauder basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Schauder_basis]`_`f for `F33f`_`[square-integrable functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square-integrable_function]`_`f on a bounded domain. As a particular example, the collection { 2 sin ⁡ ⁡ ( 2 π π n x ) ∣ ∣ n ∈ ∈ N } ∪ ∪ { 2 cos ⁡ ⁡ ( 2 π π n x ) ∣ ∣ n ∈ ∈ N } ∪ ∪ { 1 } {\\displaystyle \\{{\\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 for `F33f`_`[L2[0,1`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lp_space]`_`f.

>>See also

• `F33f`_`[Basis (linear algebra)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Basis_(linear_algebra)]`_`f (`F33f`_`[Hamel basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hamel_basis]`_`f)

• `F33f`_`[Schauder basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Schauder_basis]`_`f (in a `F33f`_`[Banach space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Banach_space]`_`f)

• `F33f`_`[Dual basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dual_basis]`_`f

• `F33f`_`[Biorthogonal system`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Biorthogonal_system]`_`f (Markushevich basis)

• `F33f`_`[Orthonormal basis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthonormal_basis]`_`f in an `F33f`_`[inner-product space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inner-product_space]`_`f

• `F33f`_`[Orthogonal polynomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthogonal_polynomials]`_`f

• `F33f`_`[Fourier analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fourier_analysis]`_`f and `F33f`_`[Fourier series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fourier_series]`_`f

• `F33f`_`[Harmonic analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harmonic_analysis]`_`f

• `F33f`_`[Orthogonal wavelet`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Orthogonal_wavelet]`_`f

• `F33f`_`[Biorthogonal wavelet`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Biorthogonal_wavelet]`_`f

• `F33f`_`[Radial basis function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Radial_basis_function]`_`f

• `F33f`_`[Finite-elements (bases)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finite_element_analysis]`_`f

• `F33f`_`[Functional analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_analysis]`_`f

• `F33f`_`[Approximation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Approximation_theory]`_`f

• `F33f`_`[Numerical analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Numerical_analysis]`_`f

>>References

• `:citerefit-1993`aItô, Kiyosi (1993). `*Encyclopedic Dictionary of Mathematics`* (2nd ed.). MIT Press. p. 1141. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-262-59020-4.

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