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Saff and Varga [1977, 1978] discovered the surprising fact that given a real valued function on [-1, 1] one can sometimes obtain a better rational approximation to ƒ of a given type by allowing complex coefficients than by restricting the coefficients to be real. In this note we point out a connection between this result and the "Trefethen effect" (Trefethen [1981a, 1981b]) of near circular error curves for best approximation on the unit disc, which as Trefethen has shown is in turn closely related to the Carathéodory-Fejér Theorem and its generalisation to meromorphic approximation due to Takagi. We consider a compact simply connected set £1 in the complex plane which is symmetric about the real line (i.e. z E fl iff F E £2). We assume further that the complement of £2 is simply connected in the extended plane and that £1 is a Faber domain (see e.g. Gaier [1980]). Finally if \p denotes the conformai mapping of the complement of the unit disc D onto the complement of Q such that i//(°°) = °°, lim^^oo i//(w)/w finite real and positive, we assume for simplicity that \p can be extended continuously to the boundary of the disc. All these properties are, of course, satisfied for the main case of interest here, £1 = 7 = [ l , l ] . 21 (£1) will denote the set of functions continuous on £1 and analytic at interior points, and 21*(£2) the subspace of functions satisfying f(z) f(z). I : 21(D) —> 21(̂ 2) is the Faber transform (Gaier [1980]); #£„(ƒ; 12) the error of best (Chebyshev) approximation to ƒ G 21^(12) from SRm„(12), the set of type (m, n) rational functions with no poles in 12, and E^n(f; 12) the corresponding error of best approximation from the subset of rational approximations with real coefficients. Saff and Varga [1977, 1978] observed that it is possible to find ƒ G 2l*(7) for whichE%n(f;I)< E*n(f\I) (e.g. ƒ(x) = x ,n= 1). Some authors attacked the case n = 1 by largely geometric arguments and showed among other things

A note on a problem of Saff and Varga concerning the degree of complex rational approximation to real valued functions