OM 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,$\phi\rightarrow 0(n\rightarrow \infty), 0<\alpha\leq2$ and $$C^{-1}\leq \phi_{n+1}/\phi_{n}\leq C (n=1,2,\ldots)$$ for some constrant $C>0$. Then for any $f\in C-2a,2a]$, $$\parallel L_{n}(f)-f\parallel_{C a,a]}= O(\phi^{\alpha}_{n})$$ inplies $f \in Lip^{*}\alpha$, where $$Lip*\alpha={f\in C-2a,2a]|\omega_{2}(f,\delta)_{-2a,2a]}=O(\delta^{\alpha})}.$$ Then some similar results on $C_{2\pi$ are given, and further some results on $C-2a,2a]$ are established by adding some proper conditions.