Orthogonal functions: Difference between revisions

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{{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 [[bilinear form#Reflexivity and orthogonalityOrthogonality_(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.

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''²-norm]]s <math display="inline"> \Vertleft\| f_n \Vertright\| _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 / \Vertleft\| f_n \Vertright\| _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 (-\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:

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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^{- \frac {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.

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==See also==

* [[Hilbert space]]

* [[Eigenvalues and eigenvectors]]

* [[WannierHilbert functionspace]]

* [[Lauricella's theorem]]

* [[Karhunen–Loève theorem]]

* [[Lauricella's theorem]]

* [[Wannier function]]

==References==