Least-squares function approximation: Difference between revisions

Revision as of 13:57, 8 June 2016

Revision as of 04:18, 12 October 2016

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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=0-486-65656-X |publisher=Dover Publications |year=1988 |edition=Reprint of 1956 Prentice–Hall |url=httphttps://books.google.com/books?id=6E85hExIqHYC&pg=PA212}}

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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''_''n'' }}||² 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=httphttps://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}}

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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=httphttps://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0-8218-4790-2 |year=2009 |publisher=American Mathematical Society }}

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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=httphttps://books.google.com/books?id=8ummgMVRev0C&pg=PA30 |isbn=0-387-97517-9 |year=1991 |edition=3rd |publisher=Springer}}

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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=httphttps://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]].