Real vs. complex rational Chebyshev approximation on an interval

Abstract:

If $f \in C[ - 1,1]$ is real-valued, let ${E^{r}(f)}$ and ${E^{c}(f)}$ be the errors in best approximation to $f$ in the supremum norm by rational functions of type $(m,n)$ with real and complex coefficients, respectively. It has recently been observed that ${E^c}(f) < {E^r}(f)$ can occur for any $n \geqslant 1$, but for no $n \geqslant 1$ is it known whether ${\gamma _{mn}} = \inf _f {E^c}(f)/{E^{r}(f)}$ is zero or strictly positive. Here we show that both are possible: ${\gamma _{01}} > 0$, but ${\gamma _{mn}} = 0$ for $n \geqslant m + 3$. Related results are obtained for approximation on regions in the plane.