Approximation of periodic functions by Fourier sums in the mean (Q1100666)

In three theorems the authors show the asymptotic equations for \({\mathcal E}_ n(X)=\sup_{f\in X}\| \rho_ n(f;x)\|_ L,\) where X is \(L^{\psi}_{\beta,1}\) or \(L^{\psi}_{\beta,1}H_{\omega_ L}\) and \(\rho_ n(f;x)=f(x)-S_{n-1}(f;x).\) \(S_ n(f;x)\) is the partial sum of order n of the Fourier series of f(x).