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Electronic Transactions on Numerical Analysis.

Volume 25, pp. 54-66, 2006.

Copyright2006, Kent State University.

ISSN 1068-9613.

ETNA

Kent State University [email protected]

A REMARK ON UNIQUENESS OF BEST RATIONAL APPROXIMANTS OF DEGREE 1 IN

L2

OF THE CIRCLE

L. BARATCHARTy

Dedicated to Ed Saff on the occasion of his 60th birthday

Abstract. We derive a criterion for uniqueness of a critical point inH2 rational approximation of degree 1. Although narrowly restricted in scope because it deals with degree 1 only, this criterion is interesting because it addresses a large class of functions. The method elaborates on the topological approach in [L. Baratchart and F. Wielonsky, Rational approximation in the real Hardy spaceH2and Stieltjes integrals: a uniqueness theorem, Constr. Approx., 9 (1993), pp. 1–21] and [L. Baratchart et al., A criterion for uniqueness of a critical points inH2 rational approximation, Canad. J. Math., 47 (1995), pp. 1121–1147].

Key words. rational approximation, uniqueness, Hardy spaces, critical points AMS subject classifications. 31A25, 30E10, 30E25, 35J05

Received September 12, 2005. Accepted for publication January 17, 2005. Recommended by D. Lubinsky.

yINRIA, BP 93, 06902 Sophia-Antipolis Cedex, France ([email protected]).

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