Numerische Mathematik

, Volume 58, Issue 1, pp 641–660 | Cite as

Tracking poles and representing Hankel operators directly from data

  • J. W. Helton
  • P. G. Spain
  • N. J. Young


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 bestL2 fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL2 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.

Subject classifications

AMS(MOS): 30-04 30E10 41A20 65E05 93-04 CR: G1.2 


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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. W. Helton
    • 1
  • P. G. Spain
    • 2
  • N. J. Young
    • 3
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of GlasgowGlasgowScotland
  3. 3.Department of MathematicsUniversity of LancasterUK

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