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In [[mathematics]], the idea of '''least squares''' can be applied
{{See also|Fourier series|Generalized Fourier series}}
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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=048665656X |publisher=Dover Publications |year=1988 |edition=Reprint of 1956
</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
{{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=0821847902 |publisher=American Mathematical Society Bookstore |year=2009 |edition=Reprint of Wadsworth and Brooks/Cole 1992}}
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{{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=0387975179 |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>
:<math>a_j = \int_a^b \phi _j^* (x)f (x) \, dx. </math>
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