Real vs. complex rational Chebyshev approximation on an interval (Q756047)

Let I denote the closed interval [-1,1], let \(\pi^ r_ m\) \((\pi^ c_ m)\) denote the set of polynomials of degree at most m with real (complex) coefficients, and let \(\pi^ r_{m,n}\) \((\pi^ c_{m,n})\) denote the set of rational functions of the form p/q, where \(p\in \pi^ r_ m\) \((\pi^ c_ m)\) and \(q\in \pi^ r_ n\) \((\pi^ c_ n)\). For \(f\in C^ r(I)\), the collection of real valued functions on I, let \(E^ r_{m,n}(f)=\inf \{\| f-g\|_ I:\;g\in \pi^ r_{m,n}\},\) and let \(E^ c_{m,n}(f)=\inf \| f-g\|_ I:\;g\in \pi^ c_{m,n}\},\) where \(\| \cdot \|_ I\) denotes the sup norm of I. For m and n positive integers, let \(\gamma_{m,n}=\inf \{E^ c_{m,n}(f)/E^ r_{m,n}(f):\;f\in C^ r(I)\}.\) \textit{L. N. Trefethen} and \textit{M. H. Gutknecht} [Trans. Am. Math. Soc. 280, 555-561 (1983; Zbl 0552.41009)] proved that \(\gamma_{m,n}=0\) whenever \(n\geq m+3\). \textit{A. Levin} [Constructive Approximation 2, 213-219 (1986; Zbl 0592.41018)] showed that \(\gamma_{m,n}=1/2\) whenever \(1\leq n\leq m+1\). The authors show that \(\gamma_{m,m+2}\leq 1/3\). The construction used here is basically different from the construction used by Trefethen and Gutknecht and by Levin.