## Discrete least squares approximation by trigonometric polynomials.(English)Zbl 0733.65102

Authors’ summary: We present an efficient and reliable algorithm for discrete least squares approximation of a real-valued function given at arbitrary distinct nodes in [0,2$$\pi$$) by trigonometric polynomials. The algorithm is based on a scheme for the solution of an inverse eigenproblem for unitary Hessenberg matrices and requires only O(mn) arithmetic operations as compared with $$O(mn^ 2)$$ operations needed for algorithms that ignore the structure of the problem.
Moreover, the proposed algorithm produces consistently accurate results that are often better than those obtained by general QR decomposition methods for the least squares problem. Our algorithm can also be used for discrete least squares approximation on the unit circle by algebraic polynomials.

### MSC:

 65T40 Numerical methods for trigonometric approximation and interpolation 65D10 Numerical smoothing, curve fitting 65F30 Other matrix algorithms (MSC2010) 42A10 Trigonometric approximation
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