The authors improve a result in \textit{D.~Gaier} [Lectures on complex approximation (1987; Zbl 0612.30003)] on the perfect convergence of the Lagrange interpolation for analytic functions on \([-1,1]\). The perfect convergence for the trigonometric interpolation of analytic functions on \([-\pi,\pi]\) with period \(2\pi\) is discussed. The trigonometric interpolation for some special analytic functions on \([-\pi, \pi]\) with period \(2\pi\) is reduced to the Lagrange interpolation for some analytic functions on \([-1,1]\), whence the perfect convergence theorem for the trigonometric interpolation of analytic functions on \([-\pi,\pi]\) with period \(2\pi\) follows.