Least-squares function approximation: Difference between revisions

Revision as of 14:00, 22 March 2013

Revision as of 08:50, 10 July 2013

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In [[mathematics]], the idea of '''least squares''' can be applied to [[function approximation|approximating a given function]] by a weighted sum of other functions. The best approximation can be defined as that which minimises the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared differences between the two.

==Functional analysis==

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:<math> \int_a^b \phi _i^* (x)\phi _j (x) \, dx =\delta_{ij},</math>

where ''δ''_''ij'''' is the [[Kronecker delta]]. Substituting function {{nowrap|''f''_''n''}} into these equations then leads to

the ''n''-dimensional [[Pythagorean theorem]]:<ref name=Wood>