For r > 0 let A P ( D r ) denote the set of 2 π -periodic functions which are analytic on the closed rectangle D r = { z ∈ C : 0 ≤ Re z ≤ 2 π , | Im z | ≤ r } , and let A P [ 0 , 2 π ] = A P ( D 0 ) . For a positive integer n let Z n = { t n , 1 , t n , 2 , … , t n , n } be a set of nodes in the interval [ 0 , 2 π ) such that t n , 1 < t n , 2 < ⋯ < t n , n , and let T n Z f denote the trigonometric interpolation operator which interpolates f ∈ A P [ 0 , 2 π ] on the set Z n . Finally, set (∗) h = lim sup n → ∞ ‖ ∏ j = 1 n sin t − t n , j 2 ‖ n , where ‖ f ‖ denotes the maximum norm of continuous 2 π -periodic function f on the interval [ 0 , 2 π ] . In this paper, we define a function r : [ 1 2 , 1 ] → [ 0 , ∞ ) by r ( h ) to be the infimum of all numbers r > 0 such that the sequence ( T n Z f ) converges to f uniformly on [ 0 , 2 π ] for every f ∈ A P ( D r ) and every sequence of nodal sets Z n which satisfy equation (∗). The main results are summarized as ln ( 4 h − 1 ) ≤ r ( h ) ≤ min { 2 ln ( h + 1 + h 2 ) , ln ( 4 h 2 + 16 h 4 − 1 ) } .