On the application of a generalized translation operator in the approximation theory (Q1406390)

The author investigates the generalized translation operator whose Fourier-Jacobi series \[ \sum\limits_{k=0}^\infty a_k\varphi_k(h)P_k^{(\alpha,\beta)}(x), \] is asymmetric. Here \(\,\{\varphi_k(h)\}^{\infty}_{k=0}\,\) is a system of functions. The author proves an analogue of the assertion: For the \(2\pi\)-periodic functions \(F\) the conditions \[ E_n(F)_{p^*} = O(n^{-r})\quad\text{and}\quad \omega(F,\delta)_{p^*} = O(\delta^r) \] are equivalent, where \(\,0<r<1\,\) and \(\,E_n(F)_{p^*}\) is the best approximation in the metric \(L_{p^*}\) of the function \(F\) by the trigonometric polynomials of the order not higher than \(n-1\), and \(\,\omega(F,\delta)_{p^*}\) is the ordinary continuity module of the function \(F\) in the metric \(L_{p^*}\).