ON AN INVERSE THEOREM OF APPROXIMATION

The author gives some disagreement to the following result,which is published in 1].Let {L_n(f(}be mass-concerntative,φ_n→0+(n→∞),0<α≤2 andC~(-1)≤(φn+1)/(φn)≤C(n=1,2,…)for some constrant C>0.Then for any f∈C-2a,2a],||L_n(f)-f||σ-a,a]=O(φ_n~α)implies f∈Lip*α,whereLip*a={f∈C-2a,2a]|w_2(f,δ)-2a,2a]=O(δ~α)}.Then some similar results on C_(2π)are given,and further some results on C-2a,2a]areestablished by adding some proper conditions.