Tracking poles and representing Hankel operators directly from data |
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Authors: | J W Helton P G Spain N J Young |
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Institution: | (1) Department of Mathematics, University of California at San Diego, 92093 La Jolla, CA, USA;(2) Department of Mathematics, University of Glasgow, G12 8QW Glasgow, Scotland;(3) Department of Mathematics, University of Lancaster, LA 1 4YF, UK |
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Abstract: | Summary 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.Partially supported by grants from the AFOSR and NSF |
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Keywords: | AMS(MOS): 30-04 30E10 41A20 65E05 93-04 CR: G1 2 |
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