Orthogonal functions: Difference between revisions

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{{Short description|Type of function}}

In [[mathematics]], two [[function (mathematics)|functions]] <math>f</math> and <math>g</math> are called '''orthogonal''' if their [[inner product]] <math>\langle f,g\rangle</math> is zero for ''f''&nbsp;≠&nbsp;''g''.

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 display="block"> \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.

==Choice of inner product==

How the [[inner product]] of two functions is defined may vary depending on context. However, a typical definition of an inner product for functions is

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]].

<math display="block"> \langle f,g\rangle = \int f(x) ^* g(x)\,dx </math>

==Trigonometric functions==

with appropriate [[integral|integration]] boundaries. Here, the asterisk indicates the [[complex conjugate]] of ''f''.

{{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==

For another perspective on this inner product, suppose approximating vectors <math>\vec{f}</math> and <math>\vec{g}</math> are created whose entries are the values of the functions ''f'' and ''g'', sampled at equally spaced points. Then this inner product between ''f'' and ''g'' can be roughly understood as the dot product between approximating vectors <math>\vec{f}</math> and <math>\vec{g}</math>, in the limit as the number of sampling points goes to infinity. Thus, roughly, two functions are orthogonal if their approximating vectors are perpendicular (under this common inner product).

*{{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]].

==Examples==

Examples of sets of orthogonal functions:

*[[Fourier series|Sines and cosines]]

*[[Bessel function]]s

*[[Hermite polynomials]]

*[[Laguerre polynomials]]

*[[Legendre polynomials]]

*[[Spherical harmonics]]

*[[Walsh function]]s

*[[Zernike polynomials]]

*[[Chebyshev polynomials]]

==See also==

* [[Hilbert space]]

* [[Harmonic analysis]]

* [[Orthogonal polynomials]]

* [[Orthonormal basis]]

* [[Eigenfunction]]

* [[Eigenvalues and eigenvectors]]

* [[Hilbert space]]

{{Unreferenced|date=January 2008}}

* [[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 ==

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[[Category:Functional analysis]]

[[Category:Types of functions]]