Asymptotics of the approximation of individual functions by generalized Fejér operators (Q1297740)

For every \(n\in \mathbb N=\{1,2,\dots\}\) let \(m\in\{2,3,\dots\}\) such that \(m(n-1)2^{-1}\) is an integer. The operators \(L_m(n)\), \(n\in\mathbb N\), are \(L_m(n,f;x)=K^{-1}_m(n)\int^\pi_{-\pi}f(x+t)(\sin nt/2)^m(\sin t/2)^{-m}\) for every \(n\in\mathbb N\) and \(f\in C_{2\pi}(\mathbb R,\mathbb R)\) where \(K_m(n)=\int^\pi_{-\pi}(\sin nt/2)^m(\sin t/2)^{-m}dt\) for every \(n\in\mathbb N\) and \(C_{2\pi}\) denotes the continuous and \(2\pi\) periodic functions. The operators \(L_m(n)\), \(n\in\mathbb N\), include the Fejér polynomials, Jackson polynomials, Fourier polynomials and some of their generalizations. In this paper complete asymptotic expansions are obtained for the estimates of the approximation of finitely differentiable functions as well as analytic functions by the above operators.