Tracking poles and representing Hankel operators directly from data

Helton, J. W., Spain, P. G. and Young, N. J. (1990) Tracking poles and representing Hankel operators directly from data. Numerische Mathematik, 58(1), pp. 641-660. (doi: 10.1007/BF01385646)

Abstract

We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL 2 fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL 2 which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH ∞, analytic, approximation toG relative to theL ∞ norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Spain, Dr Philip
• Authors: Helton, J. W., Spain, P. G., and Young, N. J.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Numerische Mathematik
• Publisher: Springer-Verlag
• ISSN: 0029-599X
• ISSN (Online): 0945-3245