Hermite interpolation problems with equally spaced nodes on the unit circle are considered. This problem is decomposed into two problems I and II. The Hermite interpolation problem of type I with equally spaced nodes on the unit circle (well-known as Hermite-Fejér interpolation problem) is said to be a problem to determine a polynomial such that and for . In Theorem 2 it is proved that a solution of the Hermite interpolation problem of type I is given by , where
The Hermite interpolation problem of type II is solved in Theorem 3. To compute the coefficients , the fast Fourier transform (FFT) is used. The general Hermite interpolation problems with an arbitrary number of derivatives in the case of algebraic and Laurent polynomials are also considered. In Section 4 algorithms for computing the Hermite interpolation polynomials in the interval based on the Tchebycheff nodes and the Hermite trigonometric interpolation problems on are considered.