Orthogonal functions: Difference between revisions

Revision as of 10:28, 13 June 2016

Revision as of 13:23, 14 June 2016

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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

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

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).[http://maze5.net/?page_id=369]

==In differential equations==

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* [[Eigenvalues and eigenvectors]]

{{Unreferenced|date=January 2008}}

== External links ==

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

[[Category:Functional analysis]]