Orthogonal functions: Difference between revisions

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{{Short description|Type of function}}

In [[mathematics]], '''orthogonal functions''' belong to a [[function space]] whichthat is a [[vector space]] (usually over R) thatequipped haswith 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 f(x) g(x)\,dx </math>

:<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.

Then functions ''f'' and ''g'' are '''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 <math> \{ f_0, f_1, \ldots\}</math> is a sequence of orthogonal functions of nonzero [[L2-norm|''L''²-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''²-norm one, forming an [[orthonormal sequence]]. To have a defined ''L''²-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 (-π\pi, π\pi),</math> when <math>m \neq n</math> ifand ''mn'' ≠and ''nm'' 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>

so thatand the integral of the product of the two sinessine 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==

When the function space consists of [[polynomial]]s, the orthogonal functions are [[orthogonal polynomials]], which have several varieties.

{{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]].

==Other sets of orthogonal functions==

*[[Bessel function]]s

*[[Hermite polynomials]]

*[[Laguerre polynomials]]

*[[Legendre polynomials]]

*[[Spherical harmonics]]

*[[Walsh function]]s

*[[Zernike polynomials]]

*[[Chebyshev polynomials]]

==See also==

* [[Hilbert space]]

* [[Harmonic analysis]]

* [[Orthonormal basis]]

* [[Eigenfunction]]

* [[Eigenvalues and eigenvectors]]

* [[Hilbert space]]

* [[Karhunen–Loève theorem]]

* [[Lauricella's theorem]]

* [[BesselWannier function]]s

==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]]