## A remark on uniqueness of best rational approximants of degree 1 in $L^2$ of the circle

L. Baratchart

### Abstract

We derive a criterion for uniqueness of a critical point in $H^2$ 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 space $H^2$ and 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 in $H^2$ rational approximation, Canad. J. Math., 47 (1995), pp. 1121–1147].

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### Key words

rational approximation, uniqueness, Hardy spaces, critical points

### AMS subject classifications

31A25, 30E10, 30E25, 35J05

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