A note on real vs. complex best Chebyshev approximation on an interval (Q1072034)

Let \(E^ R_{nn}(f)\) \((E^ C_{nn}(f))\) be the error in the best Chebyshev approximation of a real continuous function f on [-1,1] by real (complex) rational functions of type (n,n). We show that the ratio \(E^ C_{nn}(f)/E^ R_{nn}(f)\) may be arbitrarily close to 1/2 and that for the class of even functions and \(n=1\) this bound is sharp. We also prove that \(\inf \{E^ C_{11}(f)/E^ R_{11}(f):E^ R_{11}(f)>0\}\) is positive.