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Then functions ''f'' and ''g'' are [[bilinear form#Reflexivity and orthogonality|orthogonal]] when this integral is zero: <math>\langle f, \ g \rangle = 0.</math>
As with a [[basis (linear algebra)|basis]] of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.
Suppose {''f''<sub>n</sub>}, n = 0, 1, 2, … is a sequence of orthogonal functions. If ''f''<sub>n</sub> has positive [[support (mathematics)|support]] then <math>\langle f_n, f_n \rangle = \int f_n ^2 \ dx = m_n </math> is the [[L2-norm]] of ''f''<sub>n</sub>, and the sequence <math>\{ \frac {f_n}{m_n} \}</math> has functions of L2-norm one, forming an [[orthonormal sequence]]. The possibility that an integral is unbounded must be avoided, hence attention is restricted to [[square-integrable function]]s.
==Trigonometric functions==
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==Binary-valued functions==
[[Walsh function]]s and [[Haar wavelet]]s are examples of orthogonal functions with discrete ranges.
==Rational functions==
The [[rational function]] <math>\frac {x - 1}{x + 1} </math> is used as an argument in Chebyshev polynomials and Legendre polynomials to form sequences of orthogonal rational functions called [[Chebyshev rational functions]] and [[Legendre rational functions]].
==In differential equations==
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* [[Hilbert space]]
* [[Harmonic analysis]]
* [[Eigenvalues and eigenvectors]]
* [[Wannier function]]
* [[Lauricella's theorem]]
* [[Karhunen-Loeve theorem]]
==References==
{{reflist}}
* George B. Arfken & Hans J. Weber (2005) ''Mathematical Methods for Physicists'', 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, [[Academic Press]].
* [[Giovanni Sansone]] (translated by Ainsley H. Diamond) (1959) ''Orthogonal Functions'', [[Interscience Publishers]].
== External links ==
* [http://mathworld.wolfram.com/OrthogonalFunctions.html Orthogonal Functions], on MathWorld.
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