On Sliding Window Schemes For DiscreteLeast

Fast, eecient, and reliable algorithms for up-and downdating discrete least-squares approximations of a real-valued function given at arbitrary distinct nodes in 0; 2) by trigonometric polynomials are presented. A combination of the up-and downdating algorithms yields a sliding window scheme. The 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. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.