Orthogonal functions: Difference between revisions

Revision as of 00:19, 4 August 2016

Revision as of 23:09, 4 August 2016

Line 1:

In [[mathematics]], '''orthogonal functions''' belong to a [[function space]] which is a [[vector space]] (usually over R) that has 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>

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.

Line 11:

so that the integral of the product of the two sines vanishes.<ref>[[Antoni Zygmund]] (1935) ''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|Orthogonal polynomials}}

If one begins with the [[polynomial]] sequence {1, ''x'', ''x''², ... ''x''ⁿ ...} on [–1, 1] 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 a weight function ''w''(''x'') which is inserted in the bilinear form

:<math> \langle f,g\rangle = \int w(x) f(x) g(x)\,dx .</math>

For [[Laguerre polynomial]]s on (0, ∞) the weight function is <math>w(x) = e^{-x} .</math>

Both probabilists and physicists use [[Hermite polynomial]]s on (−∞, ∞) 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 [−1, 1] and use weights <math>w(x) = \frac{1}{\sqrt{1 - x^2}}</math> or <math>w(x) = \sqrt{1 - x^2}</math>

[[Zerniko 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.

==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==

Line 30 ⟶ 35:

* [[Harmonic analysis]]

* [[Orthonormal basis]]

* [[Eigenfunction]]

* [[Eigenvalues and eigenvectors]]

==References==