Improved converse theorems and fractional moduli of smoothness in Orlicz spaces (Q1941007)

Converse results of trigonometric approximation of functions and their fractional derivatives in certain Orlicz spaces are improved. More precisely, the converse inequality given by \textit{R. Akgün} and \textit{D. M. Israfilov} [Glas. Mat., III. Ser. 43, No. 1, 121--136 (2008; Zbl 1152.30038), Theroem 2] \[ \omega_{r}(f;1/(n+1))_{M}\leq \frac{C(M, r)}{(n+1)^{r}}\sum_{\mu=0}^{n}(\mu+1)^{r-1}E_{\mu}(f)_{M}, \] is improved to \[ \omega_{r}(f;1/(n+1))_{M}\leq \frac{C(M, r)}{(n+1)^{r}}\left \{\sum_{\mu=0}^{n}(\mu+1)^{rq-1}E_{\mu}^{q}(f)_{M}\right \}^{1/q}, \] for the case that \(M(u^{1/q})\) is convex for some \(1<q\leq 2\). For a definition of the appearing quantities in this estimate, see the paper.