The region of analytic functions for the convergence of trigonometric interpolation

For r>0 let AP(Dr) denote the set of 2π-periodic functions which are analytic on the closed rectangle Dr={z∈C:0≤Re z≤2π,|Im z|≤r}, and let AP[0,2π]=AP(D0). For a positive integer n let Zn={tn,1,tn,2,…,tn,n} be a set of nodes in the interval [0,2π) such that tn,10 such that the sequence (TnZf) converges to f uniformly on [0,2π] for every f∈AP(Dr) and every sequence of nodal sets Zn which satisfy equation (∗). The main results are summarized as ln(4h−1)≤r(h)≤min{2ln(h+1+h2),ln(4h2+16h4−1)}.