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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.
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==Functional analysis==
 
{{See also|Fourier series|Generalized Fourier series}}
 
A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an [[orthogonal functions|orthogonal set]]:<ref name=Lanczos>
 
{{cite book |title=Applied analysis |author=Cornelius Lanczos  |pages =212–213 |isbn=0-486-65656-X |publisher=Dover Publications |year=1988 |edition=Reprint of 1956 Prentice–Hall |url=http://books.google.com/books?id=6E85hExIqHYC&pg=PA212}}
</ref>
 
:<math>f(x) \approx f_n (x) = a_1 \phi _1 (x) + a_2 \phi _2(x) + \cdots + a_n \phi _n (x), \ </math>
 
with the set of functions {<math>\  \phi _j (x) </math>} an [[Orthonormal_set#Real-valued_functions|orthonormal set]] over the interval of interest, {{nowrap|say [a, b]}}: see also [[Fejér's theorem]]. The coefficients {<math>\ a_j </math>} are selected to make the magnitude of the difference  ||{{nowrap|''f'' − ''f''<sub>''n''</sub> }}||<sup>2</sup> as small as possible. For example, the magnitude, or norm, of a function {{nowrap|''g'' (''x'' )}} over the {{nowrap|interval [a, b]}} can be defined by:<ref name=Folland>
 
{{cite book |title=Fourier analysis and its application |page =69 |chapter=Equation 3.14 |author=Gerald B Folland |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0-8218-4790-2 |publisher=American Mathematical Society Bookstore |year=2009 |edition=Reprint of Wadsworth and Brooks/Cole 1992}}
 
</ref>
 
:<math> \|g\| = \left(\int_a^b g^*(x)g(x) \, dx \right)^{1/2} </math>
 
where the ‘*’ denotes complex conjugate in the case of complex functions. The extension of Pythagoras' theorem in this manner leads to [[function space]]s and the notion of [[Lebesgue measure]], an idea of “space” more general than the original basis of Euclidean geometry. The {{nowrap|{ <math>\phi_j (x)\ </math> } }} satisfy  [[Orthogonal#Orthogonal_functions|orthonormality relations]]:<ref name=Folland2>
{{cite book |title=Fourier Analysis and Its Applications|page =69  |first1=Gerald B | last1= Folland|url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0-8218-4790-2 |year=2009 |publisher=American Mathematical Society  }}
</ref>
 
:<math> \int_a^b \phi _i^* (x)\phi _j (x) \, dx =\delta_{ij},</math>
 
where ''δ<sub>ij</sub>'' is the [[Kronecker delta]]. Substituting function {{nowrap|''f''<sub>''n''</sub>}} into these equations then leads to
the ''n''-dimensional [[Pythagorean theorem]]:<ref name=Wood>
 
{{cite book |title=Statistical methods: the geometric approach |author= David J. Saville, Graham R. Wood |chapter=§2.5 Sum of squares |page=30 |url=http://books.google.com/books?id=8ummgMVRev0C&pg=PA30 |isbn=0-387-97517-9 |year=1991 |edition=3rd |publisher=Springer}}
</ref>
 
:<math>\|f_n\|^2 = |a_1|^2 + |a_2|^2 + \cdots + |a_n|^2. \, </math>
 
The coefficients {''a<sub>j</sub>''} making  {{nowrap begin}}||''f'' − ''f<sub>''n''</sub>||<sup>2</sup>{{nowrap end}}  as small as possible are found to be:<ref name=Lanczos/>
 
:<math>a_j = \int_a^b \phi _j^* (x)f (x) \, dx. </math>
 
The generalization of the ''n''-dimensional Pythagorean theorem to ''infinite-dimensional&thinsp;'' [[real number|real]] inner product spaces is known as [[Parseval's identity]] or Parseval's equation.<ref name=Folland3>
 
{{cite book |title=cited work |page =77 |chapter=Equation 3.22  |author=Gerald B Folland |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA77 |isbn=0-8218-4790-2 |date=2009-01-13 }}
 
</ref> Particular examples of such a representation of a function are the [[Fourier series]] and the [[generalized Fourier series]].
 
==References==
{{reflist}}
 
[[Category:Least squares]]
[[Category:Approximation theory]]

Revision as of 13:34, 6 February 2014

Now starting seeing and feeling each the reps with perfect form. Would you break at the hips? Is the chest out and head up? Perform staying upright? What does it appear as if when you power associated with the pin?

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