A necessary condition for best approximation in monotone and sign-monotone norms

Best approximation to f{hook} ε{lunate} C[a, b] by elements of an n-dimensional Tchebycheff space in monotone norms (norms defined on C[a, b] for which |f{hook}(x)| ≤ | g(x)|, a ≤ x ≤ b, implies ∥f{hook}∥ ≤ ∥g∥) is studied. It is proved that the error function has at least n zeroes in [a, b], counting twice interior zeroes with no change of sign. This result is best possible for monotone norms in general, and improves the one in [5]. The proof follows from the observation that, for any monotone norm, sgn f{hook}(x) = sgn g(x), a ≤ x ≤ b, implies ∥f{hook}- λg ∥ < ∥f{hook}∥ for λ > 0 small enough. This property is shown to characterize a class of norms wider than the class of monotone norms, namely "sign-monotone" norms defined by: |f{hook}(x)| ≤ |g(x)|, f{hook}(x) g(x) ≥ 0, a ≤ x ≤ b, implies ∥f{hook}∥ ≤ ∥g∥. It is noted that various results concerning approximation in monotone norms, are actually valid for approximation in sign-monotone norms.