** Next:** Introduction

# Downdating of Szego Polynomials
and Data Fitting Applications

** G.S. Ammar
W.B. Gragg
L. Reichel **

**January 22, 1992**

* To appear in Linear Algebra and Its Applications*

### Abstract:

*Many algorithms for polynomial least-squares approximation of a
real-valued function on a real interval determine polynomials that are
orthogonal with respect to a suitable inner product defined on this interval.
Analogously, it is convenient to compute Szego polynomials,
i.e., polynomials that are orthogonal with respect to an inner
product on the unit circle, when approximating a complex-valued function
on the unit circle in the least-squares sense.
It may also be appropriate
to determine Szego polynomials in algorithms for least-squares
approximation of real-valued periodic functions by trigonometric polynomials.
This paper is concerned with Szego polynomials that
are defined by a discrete inner product on the unit circle. We present a
scheme for downdating the Szego polynomials and given
least-squares approximant when a node is deleted from the inner product.
Our scheme uses the QR algorithm for unitary upper Hessenberg matrices.
We describe a data-fitting application that illustrates how our scheme
can be combined with the fast Fourier transform algorithm when the given
nodes are not equidistant. Application to sliding windows is discussed
also.
*

*Greg Ammar *

Tue Feb 14 12:51:53 CST 1995