Downdating of Szegö polynomials and data-fitting applications (Q1194528)

For approximating a complex-valued function on the unit circle in the least-square sense, it is convenient to compute Szegő polynomials (that is, polynomials that are orthogonal with respect to an inner product on the unit circle). The authors study Szegő polynomials that are defined by a discrete inner product on the unit circle. A connection between Szegő polynomials and unitary upper Hessenberg matrices is established. Using the \(QR\) algorithm for unitary upper Hessenberg matrices [see \textit{W. B. Gragg}, J. Comput. Appl. Math. 16, 1- 8 (1986; Zbl 0623.65041)] a scheme is presented for downdating the Szegő polynomials and given least-squares approximant when a node is deleted from the inner product. The authors show that the downdating scheme may be combined with the fast Fourier transform algorithm to achieve rapid schemes for interpolation when the nodes in the inner product are certain subsets of the set of equidistant nodes \(\exp(2\pi ij/N)\), \(j=0,\dots,N-1\). Applications to sliding windows are also discussed.