Orthogonal functions: Difference between revisions

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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:

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''.

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

Then functions ''f'' and ''g'' are '''orthogonal''' when this integral is zero: <math>\langle f, \ g \rangle = 0.</math>

==Choice of inner product==

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.

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

==Trigonometric functions==

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

{{Main|Fourier series}}

Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions, sin ''nx'', are orthogonal on the interval (-π, π), if ''m'' ≠ ''n''. For then

:<math>2 \sin mx \sin nx = \cos (m - n)x - cos (m+n) x, </math>

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]].

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

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

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).

==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 of==Other sets of orthogonal functions:==

==Examples==

Examples of sets of orthogonal functions:

*[[Fourier series|Sines and cosines]]

*[[Bessel function]]s

*[[Hermite polynomials]]

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* [[Hilbert space]]

* [[Harmonic analysis]]

* [[Orthogonal polynomials]]

* [[Orthonormal basis]]

* [[Eigenfunction]]

* [[Eigenvalues and eigenvectors]]

==References==

{{Unreferenced|date=January 2008}}

{{reflist}}

* George B. Arfken & Hans J. Weber (2005) ''Mathematical Methods for Physicists'', 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, [[Academic Press]].

== External links ==

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