For every pair of integers, where we denote by the linear space of all Laurent polynomials (-polynomials)
Suppose that , are two nondecreasing sequences of nonnegative integers such that , with , and bounded. In this paper the following result is proved:
Theorem. Let be a continuous function on and let be the unique -polynomial in satisfying
where are the roots of , being an arbitrary sequence on . Then the sequence converges to uniformly on .
This result is an extension to the unit circle of the classical Hermite-Fejér theorem about an approximation on the interval .