On approximation by polynomials and rational functions in Orlicz spaces (Q1059243)

Let \(L^*_{\phi}(\Delta)\) denote the Orlicz space corresponding to the \(\phi\) function, that is the space of the functions \(f\), Lebesgue measurable, for which \(\int_{\Delta}\phi (| f(x)| /k)dx<\infty\) for some number \(k=k(f)>0\). Let \(E_ n(f,\Delta)_{\phi}\) and \(R_ n(f,\Delta)_{\phi}\) denote the distances in the metric of \(L^*_{\phi}(\Delta)\) of the function \(f\in L^*_{\phi}(\Delta)\) from the set of algebraic polynomials of degree n and from the set of rational functions respectively. \(\omega_ r(\delta,f,\Delta)_{\phi}=\sup_{0\leq h\leq \delta}\| \sum^{r}_{i=0}(-1)^{r-i}\binom{r}{i} f(x+ih)\|_{[a,b-rh]}\) denotes the modulus of smoothness of order \(r\). One shows that if \(f\in L^*_{\phi}(\Delta)\), \(\Delta =[-1,1]\) then for any positive integer \(r\) and \(n\geq r\), we have \(R_ n(f,\Delta)_{\phi}\leq E_ n(f,\Delta)_{\phi}\leq C(r)\omega_ r(1/n,f,\Delta)_{\phi}\) where \(C(r)\) depends only on \(r\). One proves an analogous result for the functions of period \(2\pi\) and one shows that the results cannot be improved.