## A note on Newbery's algorithm for discrete least-squares approximation by trigonometric polynomials

Heike Faßbender

### Abstract

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($mn^2$) 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.

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### Key words

trigonometric approximation.

### AMS subject classifications

65D10, 42A10, 65F99.

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