Editing Orthogonal functions

{{Short description|Type of function}}
In [[mathematics]], '''orthogonal functions''' belong to a [[function space]] that is a [[vector space]] equipped with a [[bilinear form]]. When the function space has an [[interval (mathematics)|interval]] as the [[domain of a function|domain]], the bilinear form may be the [[integral]] of the product of functions over the interval:
:<math> \langle f,g\rangle = \int \overline{f(x)}g(x)\,dx  .</math>

The functions <math>f</math> and <math>g</math> are [[Orthogonality_(mathematics)|orthogonal]] when this integral is zero, i.e. <math>\langle f, \, g \rangle = 0</math> whenever <math>f \neq g</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.  Conceptually, the above integral is the equivalent of a vector [[dot product]]; two vectors are mutually independent (orthogonal) if their dot-product is zero.

Suppose <math> \{ f_0, f_1, \ldots\}</math> is a sequence of orthogonal functions of nonzero [[L2-norm|''L''<sup>2</sup>-norm]]s <math display="inline"> \left\| f_n \right\| _2 = \sqrt{\langle f_n, f_n \rangle} = \left(\int f_n ^2 \ dx \right) ^\frac{1}{2} </math>.  It follows that the sequence <math>\left\{ f_n / \left\| f_n \right\| _2 \right\}</math> is of functions of ''L''<sup>2</sup>-norm one, forming an [[orthonormal sequence]].  To have a defined ''L''<sup>2</sup>-norm, the integral must be bounded, which restricts the functions to being [[square-integrable function|square-integrable]].

==Trigonometric functions==
{{Main article|Fourier series|Harmonic analysis}}
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions {{nowrap|sin ''nx''}} and {{nowrap|sin ''mx''}} are orthogonal on the interval <math>x \in (-\pi, \pi)</math> when <math>m \neq n</math> and ''n'' and ''m'' are positive integers. For then 
:<math>2 \sin \left(mx\right) \sin \left(nx\right) = \cos \left(\left(m - n\right)x\right) - \cos\left(\left(m+n\right) x\right), </math>
and the integral of the product of the two sine functions vanishes.<ref>[[Antoni Zygmund]] (1935) ''[[Trigonometric Series|Trigonometrical Series]]'', page 6, Mathematical Seminar, University of Warsaw</ref> Together with cosine functions, these orthogonal functions may be assembled into a [[trigonometric polynomial]] to approximate a given function on the interval with its [[Fourier series]].

==Polynomials==
{{main article|Orthogonal polynomials}}
If one begins with the [[monomial]] sequence <math> \left\{1, x, x^2, \dots\right\} </math> on the interval <math>[-1,1]</math> and applies the [[Gram–Schmidt process]], then one obtains the [[Legendre polynomial]]s. Another collection of orthogonal polynomials are the [[associated Legendre polynomials]].

The study of orthogonal polynomials involves [[weight function]]s <math>w(x)</math> that are inserted in the bilinear form:
:<math> \langle f,g\rangle = \int w(x) f(x) g(x)\,dx  .</math>
For [[Laguerre polynomial]]s on <math>(0,\infty)</math> the weight function is <math>w(x) = e^{-x}</math>.

Both physicists and probability theorists use [[Hermite polynomial]]s on <math>(-\infty,\infty)</math>, where the weight function is <math>w(x) = e^{-x^2}</math> or <math>w(x) = e^{- x^2/2}</math>.

[[Chebyshev polynomial]]s are defined on <math>[-1,1]</math> and use weights <math display="inline">w(x) = \frac{1}{\sqrt{1 - x^2}}</math> or <math display="inline">w(x) = \sqrt{1 - x^2}</math>.

[[Zernike polynomial]]s are defined on the [[unit disk]] and have orthogonality of both radial and angular parts.

==Binary-valued functions==
[[Walsh function]]s and [[Haar wavelet]]s are examples of orthogonal functions with discrete ranges.

==Rational functions==
[[File:ChebychevRational1.png|thumb|Plot of the Chebyshev rational functions of order n=0,1,2,3 and 4 between x=0.01 and 100.]]
Legendre and Chebyshev polynomials provide orthogonal families for the interval {{nowrap|[−1, 1]}} while occasionally orthogonal families are required on {{nowrap|[0, ∞)}}. In this case it is convenient to apply the [[Cayley transform#Real homography|Cayley transform]] first, to bring the argument into {{nowrap|[−1, 1]}}. This procedure results in families of [[rational function|rational]] orthogonal functions called [[Legendre rational functions]] and [[Chebyshev rational functions]].

==In differential equations==
Solutions of linear [[differential equation]]s with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. [[eigenfunction]]s), leading to [[generalized Fourier series]].

==See also==
* [[Eigenvalues and eigenvectors]]
* [[Hilbert space]]
* [[Karhunen–Loève theorem]]
* [[Lauricella's theorem]]
* [[Wannier function]]

==References==
{{reflist}}
* George B. Arfken & Hans J. Weber (2005) ''Mathematical Methods for Physicists'', 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, [[Academic Press]].
* {{cite journal|author=Price, Justin J.|authorlink=Justin Jesse Price|title=Topics in orthogonal functions|journal=[[American Mathematical Monthly]]|volume=82|year=1975|pages=594–609|url=http://www.maa.org/programs/maa-awards/writing-awards/topics-in-orthogonal-functions|doi=10.2307/2319690}}
* [[Giovanni Sansone]]  (translated by Ainsley H. Diamond) (1959) ''Orthogonal Functions'', [[Interscience Publishers]].

== External links ==
* [http://mathworld.wolfram.com/OrthogonalFunctions.html Orthogonal Functions], on MathWorld.

[[Category:Functional analysis]]
[[Category:Types of functions]]