Orthogonal functions: Difference between revisions

Revision as of 23:16, 4 August 2016

Revision as of 06:57, 5 August 2016

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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.

==Trigonometric functions==

{{Main article|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]].

==Polynomials==

{{main article|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]].

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* [[Orthonormal basis]]

* [[Eigenvalues and eigenvectors]]

==References==

{{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.