Geodesic
A note on Newbery's algorithm for discrete least-squares approximation by trigonometric polynomials
Electronic transactions on numerical analysis, Tome 4 (1996), pp. 64-74.

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Summary: Recently fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in [0, $2\pi $) by trigonometric polynomials were presented in different papers. These algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only $O(mn)$ arithmetic operations as compared to $O(mn2)$ operations needed for algorithms that ignore the structure of the problem. In 1970 Newbery already presented a $O(mn)$ algorithm for solving the discrete least-squares approximation by trigonometric polynomials. In this paper the connection between the different algorithms is illustrated.
Classification : 65D10, 42A10, 65F99
Keywords: trigonometric approximation 
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     author = {Fa{\ss}bender, Heike},
     title = {A note on {Newbery's} algorithm for discrete least-squares approximation by trigonometric polynomials},
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     pages = {64--74},
     publisher = {mathdoc},
     volume = {4},
     year = {1996},
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     url = {http://geodesic.mathdoc.fr/item/ETNA_1996__4__a6/}
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Faßbender, Heike. A note on Newbery's algorithm for discrete least-squares approximation by trigonometric polynomials. Electronic transactions on numerical analysis, Tome 4 (1996), pp. 64-74. http://geodesic.mathdoc.fr/item/ETNA_1996__4__a6/