A note on real vs complex best Chebyshev approximation on an interval

Abstract

Let E nn R (ƒ) (E nn C (ƒ)) be the error in the best Chebyshev approximation of a real continuous function ƒ on [−1, 1] by real (complex) rational functions of type ( n, n). We show that the ratio E nn C (ƒ) E nn R (ƒ) 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 11 C (ƒ) E 11 R (ƒ) : E 11 R (ƒ) > 0} is positive.